{
{
Short
description|Relativity
concept
expressed
as
E
{
{
=
}
}
mc²
}
}
{
{
redirect2|E
{
{
=
}
}
MC²|E
{
{
=
}
}
mc²
}
}
[
[
File
:
M87
jet.jpg|thumb|Mass
near
the
[
[
M87*
]
]
black
hole
is
converted
into
a
very
energetic
[
[
astrophysical
jet
]
]
,
stretching
five
thousand
[
[
Light-year|light
years
]
]
]
]
In
[
[
physics
]
]
,
``
'mass–energy
equivalence
''
'
is
the
relationship
between
[
[
mass
]
]
and
[
[
energy
]
]
in
a
system
's
[
[
rest
frame
]
]
,
where
the
two
quantities
differ
only
by
a
multiplicative
constant
and
the
units
of
measurement.
<
ref
name=Serway1217
>
{
{
Cite
book|last1=Serway
|first1=Raymond
A.|title=Physics
for
scientists
and
engineers
with
modern
physics|last2=Jewett
|first2=John
W.
|last3=Peroomian
|first3=Vahé|date=5
March
2013|isbn=978-1-133-95405-7|edition=9th|location=Boston
,
MA|oclc=802321453|pages=1217–1218
}
}
<
/ref
>
<
ref
name=Günther
>
{
{
Citation|last1=Günther|first1=Helmut|title=Einstein
's
Energy–Mass
Equivalence|date=2019|url=https
:
//doi.org/10.1007/978f=The
Special
Theory
of
Relativity
:
Einstein
’
s
World
in
New
Axiomatics|pages=97–105|editor-last=Günther|editor-first=Helmut|place=Singapore|publisher=Springer|language=en|doi=10.1007/978-981-13-7783-9_7|isbn=978-981-13-7783-9|access-date=2020-10-14|last2=Müller|first2=Volker|s2cid=209978258|editor2-last=Müller|editor2-first=Volker|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080229/https
:
//link.springer.com/chapter/10.1007
%
2F978-981-13-7783-9_7|url-status=live
}
}
<
/ref
>
The
principle
is
described
by
the
physicist
[
[
Albert
Einstein
]
]
's
formula
:
&
nbsp
;
<
math
qid=Q35875
>
E
=
mc^2
<
/math
>
.
<
ref
name=
''
famous
''
>
{
{
cite
book
|title=E=mc
<
sup
>
12
!
<
/sup
>
:
A
Biography
of
the
World
's
Most
Famous
Equation
|edition=illustrated
|first1=David
|last1=Bodanis
|publisher=Bloomsbury
Publishing
|year=2009
|isbn=978-0-8027-1821-1|at=preface|url=https
:
//books.google.com/books
?
id=8TX2tFLZ7gYC
}
}
<
/ref
>
In
a
[
[
reference
frame
]
]
where
the
system
is
moving
,
its
[
[
relativistic
energy
]
]
and
[
[
Mass
in
special
relativity|relativistic
mass
]
]
(
instead
of
[
[
Rest
Mass|rest
mass
]
]
)
obey
the
same
formula
.
The
formula
defines
the
energy
{
{
math|
''
E
''
}
}
of
a
particle
in
its
rest
frame
as
the
product
of
mass
(
{
{
math|
''
m
''
}
}
)
with
the
[
[
speed
of
light
]
]
squared
(
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
)
.
Because
the
speed
of
light
is
a
large
number
in
everyday
units
(
approximately
{
{
cvt|300000|km/s|mi/s|comma=gaps|sigfig=3|disp=x|
or
}
}
)
,
the
formula
implies
that
a
small
amount
of
``
rest
mass
''
,
measured
when
the
system
is
at
rest
,
corresponds
to
an
enormous
amount
of
energy
,
which
is
independent
of
the
composition
of
the
[
[
matter
]
]
.
Rest
mass
,
also
called
[
[
invariant
mass
]
]
,
is
a
fundamental
[
[
physical
property
]
]
that
is
independent
of
[
[
momentum
]
]
,
even
at
extreme
speeds
approaching
the
speed
of
light
.
Its
value
is
the
same
in
all
[
[
inertial
frame
of
reference|inertial
frames
of
reference
]
]
.
[
[
Massless
particle
]
]
s
such
as
[
[
photon
]
]
s
have
zero
invariant
mass
,
but
massless
[
[
free
particle
]
]
s
have
both
momentum
and
energy
.
The
equivalence
principle
implies
that
when
energy
is
lost
in
[
[
chemical
reaction
]
]
s
,
[
[
nuclear
reaction
]
]
s
,
and
other
[
[
energy
transformation
]
]
s
,
the
[
[
Physical
system|system
]
]
will
also
lose
a
corresponding
amount
of
mass
.
The
energy
,
and
mass
,
can
be
released
to
the
environment
as
[
[
radiant
energy
]
]
,
such
as
[
[
light
]
]
,
or
as
[
[
thermal
energy
]
]
.
The
principle
is
fundamental
to
many
fields
of
physics
,
including
[
[
nuclear
physics|nuclear
]
]
and
[
[
particle
physics
]
]
.
Mass–energy
equivalence
arose
from
[
[
special
relativity
]
]
as
a
[
[
paradox
]
]
described
by
the
French
[
[
polymath
]
]
[
[
Henri
Poincaré
]
]
(
1854–1912
)
.
<
ref
name=action
>
{
{
Cite
journal|
author=Poincaré
,
H.
|
year=1900
|
title=La
théorie
de
Lorentz
et
le
principe
de
réaction
|
journal=Archives
Néerlandaises
des
Sciences
Exactes
et
Naturelles
|
volume
=5
|
pages
=252–278|
title-link=s
:
fr
:
La
théorie
de
Lorentz
et
le
principe
de
réaction|language=fr|trans-title=
[
http
:
//physicsinsights.org/poincare-1900.pdf
The
Theory
of
Lorentz
and
The
Principle
of
Reaction
]
}
}
<
/ref
>
Einstein
was
the
first
to
propose
the
equivalence
of
mass
and
energy
as
a
general
principle
and
a
consequence
of
the
[
[
Spacetime
symmetries|symmetries
of
space
and
time
]
]
.
The
principle
first
appeared
in
``
Does
the
inertia
of
a
body
depend
upon
its
energy-content
?
``
,
one
of
his
[
[
annus
mirabilis
papers|
''
annus
mirabilis
''
papers
]
]
,
published
on
21
November
1905.
<
ref
name=
''
inertia
''
>
{
{
Cite
journal|last=Einstein|first=A.|date=1905|title=Ist
die
Trägheit
eines
Körpers
von
seinem
Energieinhalt
abhängig
?
|journal=Annalen
der
Physik|language=de|volume=323|issue=13|pages=639–641|doi=10.1002/andp.19053231314|bibcode=1905AnP
...
323..639E|trans-title=
[
http
:
//www.fourmilab.ch/etexts/einstein/E_mc2/www/
Does
the
Inertia
of
a
Body
Depend
Upon
its
Energy-Content
?
]
|issn=1521-3889|doi-access=free
}
}
<
/ref
>
The
formula
and
its
relationship
to
momentum
,
as
described
by
the
[
[
energy–momentum
relation
]
]
,
were
later
developed
by
other
physicists
.
==Description==
{
{
Special
relativity
sidebar
}
}
Mass–energy
equivalence
states
that
all
objects
having
[
[
mass
]
]
,
or
``
massive
objects
''
,
have
a
corresponding
intrinsic
energy
,
even
when
they
are
stationary
.
In
the
[
[
rest
frame
]
]
of
an
object
,
where
by
definition
it
is
motionless
and
so
has
no
[
[
momentum
]
]
,
the
mass
and
energy
are
equal
or
they
differ
only
by
a
constant
factor
,
the
[
[
speed
of
light
]
]
squared
(
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
)
.
<
ref
name=Serway1217
/
>
<
ref
name=Günther
/
>
In
[
[
Newtonian
mechanics
]
]
,
a
motionless
body
has
no
[
[
kinetic
energy
]
]
,
and
it
may
or
may
not
have
other
amounts
of
internal
stored
energy
,
like
[
[
chemical
energy
]
]
or
[
[
thermal
energy
]
]
,
in
addition
to
any
[
[
potential
energy
]
]
it
may
have
from
its
position
in
a
[
[
field
(
physics
)
|field
of
force
]
]
.
These
energies
tend
to
be
much
smaller
than
the
mass
of
the
object
multiplied
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
,
which
is
on
the
order
of
10
<
sup
>
17
<
/sup
>
&
nbsp
;
[
[
joule
]
]
s
for
a
mass
of
one
kilogram
.
Due
to
this
principle
,
the
mass
of
the
atoms
that
come
out
of
a
[
[
nuclear
reaction
]
]
is
less
than
the
mass
of
the
atoms
that
go
in
,
and
the
difference
in
mass
shows
up
as
heat
and
light
with
the
same
equivalent
energy
as
the
difference
.
In
analyzing
these
explosions
,
Einstein
's
formula
can
be
used
with
{
{
mvar|E
}
}
as
the
energy
released
(
removed
)
,
and
{
{
mvar|m
}
}
as
the
change
in
mass
.
In
[
[
Theory
of
relativity|relativity
]
]
,
all
the
energy
that
moves
with
an
object
(
i.e.
,
the
energy
as
measured
in
the
object
's
rest
frame
)
contributes
to
the
total
mass
of
the
body
,
which
measures
how
much
it
resists
[
[
acceleration
]
]
.
If
an
isolated
box
of
ideal
mirrors
could
contain
light
,
the
individually
massless
photons
would
contribute
to
the
total
mass
of
the
box
by
the
amount
equal
to
their
energy
divided
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
.
<
ref
>
{
{
Cite
book|last1=Puri|first1=H
.
S.|last2=Hans|first2=S
.
P.|url=https
:
//books.google.com/books
?
id=hrBe52GPHrYC|title=Mechanics
,
2E|date=2003-07-01|publisher=Tata
McGraw-Hill
Education|isbn=978-0-07-047360-7|language=en|page=
[
https
:
//books.google.com/books
?
id=hrBe52GPHrYC
&
pg=PA433
433
]
}
}
<
/ref
>
For
an
observer
in
the
rest
frame
,
removing
energy
is
the
same
as
removing
mass
and
the
formula
{
{
math|1=
''
m
''
=
``
E
''
/
''
c
''
<
sup
>
2
<
/sup
>
}
}
indicates
how
much
mass
is
lost
when
energy
is
removed.
<
ref
>
{
{
Cite
book|last=Serway
,
Raymond
A.|url=https
:
//www.worldcat.org/oclc/802321453|title=Physics
for
scientists
and
engineers
with
modern
physics.|others=Jewett
,
John
W.
,
Peroomian
,
Vahé.|date=5
March
2013|isbn=978-1-133-95405-7|edition=Ninth|location=Boston
,
MA|oclc=802321453|page=1386
}
}
<
/ref
>
In
the
same
way
,
when
any
energy
is
added
to
an
isolated
system
,
the
increase
in
the
mass
is
equal
to
the
added
energy
divided
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
.
<
ref
name=griffithsElectro512
>
{
{
Cite
book|last=Griffiths
,
David
J.|url=https
:
//www.worldcat.org/oclc/40251748|title=Introduction
to
electrodynamics|date=1999|publisher=Prentice
Hall|isbn=978-0-13-805326-0|edition=3rd|location=Upper
Saddle
River
,
N.J.|oclc=40251748|page=512|access-date=2020-10-15|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080229/https
:
//www.worldcat.org/title/introduction-to-electrodynamics/oclc/40251748|url-status=live
}
}
<
/ref
>
==Mass
in
special
relativity==
{
{
main|Mass
in
special
relativity
}
}
[
[
File
:
E=mc²-explication.svg|thumb|
{
{
math|1=
''
E
''
=
``
mc
''
{
{
smallsup|2
}
}
}
}
—In
[
[
SI
units
]
]
,
the
energy
{
{
math|
''
E
''
}
}
is
measured
in
[
[
Joules
]
]
,
the
mass
{
{
math|
''
m
''
}
}
is
measured
in
[
[
kilograms
]
]
,
and
the
[
[
speed
of
light
]
]
is
measured
in
[
[
meters
]
]
per
[
[
second
]
]
.
]
]
An
object
moves
at
different
speeds
in
different
[
[
Frame
of
reference|frames
of
reference
]
]
,
depending
on
the
motion
of
the
observer
.
This
implies
the
kinetic
energy
,
in
both
Newtonian
mechanics
and
relativity
,
is
'frame
dependent
'
,
so
that
the
amount
of
relativistic
energy
that
an
object
is
measured
to
have
depends
on
the
observer
.
The
``
relativistic
mass
''
of
an
object
is
given
by
the
relativistic
energy
divided
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
.
<
ref
name=
''
Tipler
''
>
{
{
Cite
book|last1=Tipler|first1=Paul
Allen|last2=Llewellyn|first2=Ralph
A.|url=
https
:
//www.worldcat.org/oclc/49894577|title=Modern
physics.|date=2003|publisher=W.H
.
Freeman|isbn=978-0-7167-4345-3|edition=4th|location=New
York|oclc=49894577|pages=87–88
}
}
<
/ref
>
Because
the
relativistic
mass
is
exactly
proportional
to
the
relativistic
energy
,
relativistic
mass
and
relativistic
energy
are
nearly
[
[
synonym
]
]
ous
;
the
only
difference
between
them
is
the
[
[
unit
of
measurement|units
]
]
.
The
``
rest
mass
''
or
[
[
invariant
mass
]
]
of
an
object
is
defined
as
the
mass
an
object
has
in
its
rest
frame
,
when
it
is
not
moving
with
respect
to
the
observer
.
Physicists
typically
use
the
term
``
mass
''
,
though
experiments
have
shown
an
object
's
gravitational
mass
depends
on
its
total
energy
and
not
just
its
rest
mass
.
{
{
Citation
needed|date=February
2021|reason=experiments
which
specifically
address
this
,
unless
what
was
meant
was
that
the
gravitational
mass
of
a
system
is
the
invariant
mass
of
the
system
as
opposed
to
the
sum
of
the
component
invariant
masses
}
}
The
rest
mass
is
the
same
for
all
[
[
inertial
frame
]
]
s
,
as
it
is
independent
of
the
motion
of
the
observer
,
it
is
the
smallest
possible
value
of
the
relativistic
mass
of
the
object
.
Because
of
the
attraction
between
components
of
a
system
,
which
results
in
potential
energy
,
the
rest
mass
is
almost
never
[
[
Additive
function|additive
]
]
;
in
general
,
the
mass
of
an
object
is
not
the
sum
of
the
masses
of
its
parts.
<
ref
name=
''
griffithsElectro512
''
/
>
The
rest
mass
of
an
object
is
the
total
energy
of
all
the
parts
,
including
kinetic
energy
,
as
observed
from
the
center
of
momentum
frame
,
and
potential
energy
.
The
masses
add
up
only
if
the
constituents
are
at
rest
(
as
observed
from
the
center
of
momentum
frame
)
and
do
not
attract
or
repel
,
so
that
they
do
not
have
any
extra
kinetic
or
potential
energy.
<
ref
group=
''
note
''
>
They
can
also
have
a
positive
kinetic
energy
and
a
negative
potential
energy
that
exactly
cancels.
<
/ref
>
Massless
particles
are
particles
with
no
rest
mass
,
and
therefore
have
no
intrinsic
energy
;
their
energy
is
due
only
to
their
momentum
.
===Relativistic
mass===
Relativistic
mass
depends
on
the
motion
of
the
object
,
so
that
different
observers
in
relative
motion
see
different
values
for
it
.
The
relativistic
mass
of
a
moving
object
is
larger
than
the
relativistic
mass
of
an
object
at
rest
,
because
a
moving
object
has
kinetic
energy
.
If
the
object
moves
slowly
,
the
relativistic
mass
is
nearly
equal
to
the
[
[
rest
mass
]
]
and
both
are
nearly
equal
to
the
classical
inertial
mass
(
as
it
appears
in
[
[
Newton
's
laws
of
motion
]
]
)
.
If
the
object
moves
quickly
,
the
relativistic
mass
is
greater
than
the
rest
mass
by
an
amount
equal
to
the
mass
associated
with
the
kinetic
energy
of
the
object
.
Massless
particles
also
have
relativistic
mass
derived
from
their
kinetic
energy
,
equal
to
their
relativistic
energy
divided
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
,
or
{
{
math|1=
''
m
''
{
{
ssub|rel
}
}
=
``
E
''
/
''
c
''
<
sup
>
2
<
/sup
>
}
}
.
<
ref
>
{
{
Cite
book|last=Mould|first=Richard
A.|url=https
:
//books.google.com/books
?
id=lfGE-wyJYIUC|title=Basic
Relativity|date=2001-11-01|publisher=Springer
Science
&
Business
Media|isbn=978-0-387-95210-9|language=en|page=
[
https
:
//books.google.com/books
?
id=lfGE-wyJYIUC
&
pg=PA126
126
]
}
}
<
/ref
>
<
ref
>
{
{
Cite
book|last=Chow|first=Tai
L.|url=https
:
//books.google.com/books
?
id=dpnpMhw1zo8C|title=Introduction
to
Electromagnetic
Theory
:
A
Modern
Perspective|date=2006|publisher=Jones
&
Bartlett
Learning|isbn=978-0-7637-3827-3|language=en|page=
[
https
:
//books.google.com/books
?
id=dpnpMhw1zo8C
&
pg=PA392
392
]
|access-date=2016-02-22|archive-date=2016-12-02|archive-url=https
:
//web.archive.org/web/20161202172249/https
:
//books.google.com/books
?
id=dpnpMhw1zo8C|url-status=live
}
}
<
/ref
>
The
speed
of
light
is
one
in
a
system
where
length
and
time
are
measured
in
[
[
natural
units
]
]
and
the
relativistic
mass
and
energy
would
be
equal
in
value
and
dimension
.
As
it
is
just
another
name
for
the
energy
,
the
use
of
the
term
``
relativistic
mass
''
is
redundant
and
physicists
generally
reserve
``
mass
''
to
refer
to
rest
mass
,
or
invariant
mass
,
as
opposed
to
relativistic
mass.
<
ref
name=elementaryParticles
>
{
{
Cite
book|last=Griffiths
,
David
J.|title=Introduction
to
elementary
particles|date=2008|publisher=Wiley-VCH|isbn=978-3-527-40601-2|edition=2nd
,
rev.|location=Weinheim
[
Germany
]
|oclc=248969635|page=101
}
}
<
/ref
>
<
ref
name=serway
>
{
{
Cite
book|last=Serway
,
Raymond
A.|title=Physics
for
scientists
and
engineers
with
modern
physics.|others=Jewett
,
John
W.
,
Peroomian
,
Vahé.|date=5
March
2013|isbn=978-1-133-95405-7|edition=Ninth|location=Boston
,
MA|oclc=802321453|page=1219
}
}
<
/ref
>
A
consequence
of
this
terminology
is
that
the
[
[
conservation
of
mass|mass
is
not
conserved
]
]
in
special
relativity
,
whereas
[
[
Momentum
#
Conservation|the
conservation
of
momentum
]
]
and
[
[
conservation
of
energy
]
]
are
both
fundamental
laws.
<
ref
name=elementaryParticles
/
>
===Conservation
of
mass
and
energy===
{
{
Main|Conservation
of
energy|Conservation
of
mass
}
}
The
conservation
of
energy
is
a
universal
principle
in
physics
and
holds
for
any
interaction
,
along
with
the
conservation
of
momentum.
<
ref
name=elementaryParticles
/
>
The
classical
conservation
of
mass
,
in
contrast
,
is
violated
in
certain
relativistic
settings.
<
ref
name=serway
/
>
<
ref
name=elementaryParticles
/
>
This
concept
has
been
experimentally
proven
in
a
number
of
ways
,
including
the
conversion
of
mass
into
kinetic
energy
in
nuclear
reactions
and
other
interactions
between
[
[
elementary
particle
]
]
s.
<
ref
name=serway
/
>
While
modern
physics
has
discarded
the
expression
'conservation
of
mass
'
,
in
older
terminology
a
[
[
relativistic
mass
]
]
can
also
be
defined
to
be
equivalent
to
the
energy
of
a
moving
system
,
allowing
for
a
``
conservation
of
relativistic
mass
''
.
<
ref
name=elementaryParticles
/
>
Mass
conservation
breaks
down
when
the
energy
associated
with
the
mass
of
a
particle
is
converted
into
other
forms
of
energy
,
such
as
kinetic
energy
,
thermal
energy
,
or
[
[
radiant
energy
]
]
.
Similarly
,
kinetic
or
radiant
energy
can
be
used
to
create
particles
that
have
mass
,
always
conserving
the
total
energy
and
momentum.
<
ref
name=elementaryParticles
/
>
===Massless
particles===
Massless
particles
have
zero
rest
mass
.
The
[
[
Planck–Einstein
relation
]
]
for
the
energy
for
[
[
photon
]
]
s
is
given
by
the
equation
{
{
math|1=
''
E
''
=
``
hf
''
}
}
,
where
{
{
mvar|h
}
}
is
the
[
[
Planck
constant
]
]
and
{
{
mvar|f
}
}
is
the
photon
[
[
frequency
]
]
.
This
frequency
and
thus
the
relativistic
energy
are
frame-dependent
.
If
an
observer
runs
away
from
a
photon
in
the
direction
the
photon
travels
from
a
source
,
and
it
catches
up
with
the
observer
,
the
observer
sees
it
as
having
less
energy
than
it
had
at
the
source
.
The
faster
the
observer
is
traveling
with
regard
to
the
source
when
the
photon
catches
up
,
the
less
energy
the
photon
would
be
seen
to
have
.
As
an
observer
approaches
the
speed
of
light
with
regard
to
the
source
,
the
[
[
redshift
]
]
of
the
photon
increases
,
according
to
the
[
[
relativistic
Doppler
effect
]
]
.
The
energy
of
the
photon
is
reduced
and
as
the
wavelength
becomes
arbitrarily
large
,
the
photon
's
energy
approaches
zero
,
due
to
the
massless
nature
of
photons
,
which
does
not
permit
any
intrinsic
energy
.
===Composite
systems===
{
{
see
also|Mass
in
special
relativity
#
The
mass
of
composite
systems
}
}
For
closed
systems
made
up
of
many
parts
,
like
an
[
[
atomic
nucleus
]
]
,
planet
,
or
star
,
the
relativistic
energy
is
given
by
the
sum
of
the
relativistic
energies
of
each
of
the
parts
,
because
energies
are
additive
in
these
systems
.
If
a
system
is
[
[
Binding
energy
#
Mass-energy
relation|
''
bound
''
]
]
by
attractive
forces
,
and
the
energy
gained
in
excess
of
the
work
done
is
removed
from
the
system
,
then
mass
is
lost
with
this
removed
energy
.
The
mass
of
an
atomic
nucleus
is
less
than
the
total
mass
of
the
[
[
proton
]
]
s
and
[
[
neutron
]
]
s
that
make
it
up.
<
ref
name=Serway1386
>
{
{
Cite
book|last=Serway
,
Raymond
A.|url=https
:
//www.worldcat.org/oclc/802321453|title=Physics
for
scientists
and
engineers
with
modern
physics.|others=Jewett
,
John
W.
,
Peroomian
,
Vahé.|date=5
March
2013|isbn=978-1-133-95405-7|edition=Ninth|location=Boston
,
MA|oclc=802321453|page=1386|access-date=15
October
2020|archive-date=21
February
2021|archive-url=https
:
//web.archive.org/web/20210221080236/https
:
//www.worldcat.org/title/physics-for-scientists-and-engineers-with-modern-physics/oclc/802321453|url-status=live
}
}
<
/ref
>
This
mass
decrease
is
also
equivalent
to
the
energy
required
to
break
up
the
nucleus
into
individual
protons
and
neutrons
.
This
effect
can
be
understood
by
looking
at
the
potential
energy
of
the
individual
components
.
The
individual
particles
have
a
force
attracting
them
together
,
and
forcing
them
apart
increases
the
potential
energy
of
the
particles
in
the
same
way
that
lifting
an
object
up
on
earth
does
.
This
energy
is
equal
to
the
work
required
to
split
the
particles
apart
.
The
mass
of
the
[
[
Solar
System
]
]
is
slightly
less
than
the
sum
of
its
individual
masses
.
For
an
isolated
system
of
particles
moving
in
different
directions
,
the
invariant
mass
of
the
system
is
the
analog
of
the
rest
mass
,
and
is
the
same
for
all
observers
,
even
those
in
relative
motion
.
It
is
defined
as
the
total
energy
(
divided
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
)
in
the
[
[
center
of
momentum
frame
]
]
.
The
``
center
of
momentum
frame
''
is
defined
so
that
the
system
has
zero
total
momentum
;
the
term
[
[
center
of
mass
]
]
frame
is
also
sometimes
used
,
where
the
``
center
of
mass
frame
''
is
a
special
case
of
the
center
of
momentum
frame
where
the
center
of
mass
is
put
at
the
origin
.
A
simple
example
of
an
object
with
moving
parts
but
zero
total
momentum
is
a
container
of
gas
.
In
this
case
,
the
mass
of
the
container
is
given
by
its
total
energy
(
including
the
kinetic
energy
of
the
gas
molecules
)
,
since
the
system
's
total
energy
and
invariant
mass
are
the
same
in
any
reference
frame
where
the
momentum
is
zero
,
and
such
a
reference
frame
is
also
the
only
frame
in
which
the
object
can
be
weighed
.
In
a
similar
way
,
the
theory
of
special
relativity
posits
that
the
thermal
energy
in
all
objects
,
including
solids
,
contributes
to
their
total
masses
,
even
though
this
energy
is
present
as
the
kinetic
and
potential
energies
of
the
atoms
in
the
object
,
and
it
(
in
a
similar
way
to
the
gas
)
is
not
seen
in
the
rest
masses
of
the
atoms
that
make
up
the
object.
<
ref
name=griffithsElectro512
/
>
Similarly
,
even
photons
,
if
trapped
in
an
isolated
container
,
would
contribute
their
energy
to
the
mass
of
the
container
.
Such
extra
mass
,
in
theory
,
could
be
weighed
in
the
same
way
as
any
other
type
of
rest
mass
,
even
though
individually
photons
have
no
rest
mass
.
The
property
that
trapped
energy
in
any
form
adds
weighable
mass
to
systems
that
have
no
net
momentum
is
one
of
the
consequences
of
relativity
.
It
has
no
counterpart
in
classical
Newtonian
physics
,
where
energy
never
exhibits
weighable
mass.
<
ref
name=griffithsElectro512
/
>
===Relation
to
gravity===
Physics
has
two
concepts
of
mass
,
the
gravitational
mass
and
the
inertial
mass
.
The
gravitational
mass
is
the
quantity
that
determines
the
strength
of
the
[
[
gravitational
field
]
]
generated
by
an
object
,
as
well
as
the
gravitational
force
acting
on
the
object
when
it
is
immersed
in
a
gravitational
field
produced
by
other
bodies
.
The
inertial
mass
,
on
the
other
hand
,
quantifies
how
much
an
object
accelerates
if
a
given
force
is
applied
to
it
.
The
mass–energy
equivalence
in
special
relativity
refers
to
the
inertial
mass
.
However
,
already
in
the
context
of
Newton
gravity
,
the
weak
[
[
equivalence
principle
]
]
is
postulated
:
the
gravitational
and
the
inertial
mass
of
every
object
are
the
same
.
Thus
,
the
mass–energy
equivalence
,
combined
with
the
weak
equivalence
principle
,
results
in
the
prediction
that
all
forms
of
energy
contribute
to
the
gravitational
field
generated
by
an
object
.
This
observation
is
one
of
the
pillars
of
the
[
[
general
theory
of
relativity
]
]
.
The
prediction
that
all
forms
of
energy
interact
gravitationally
has
been
subject
to
experimental
tests
.
One
of
the
first
observations
testing
this
prediction
,
called
the
[
[
Eddington
experiment
]
]
,
was
made
during
the
[
[
Solar
eclipse
of
May
29
,
1919
]
]
.
<
ref
>
{
{
Cite
journal|last1=Dyson|first1=F.W.|author2=Eddington
,
A.S.|author3=Davidson
,
C.R.|name-list-style=amp|date=January
1920|title=IX
.
A
determination
of
the
deflection
of
light
by
the
sun
's
gravitational
field
,
from
observations
made
at
the
total
eclipse
of
May
29
,
1919|journal=Philosophical
Transactions
of
the
Royal
Society
of
London
.
Series
A
,
Containing
Papers
of
a
Mathematical
or
Physical
Character|language=en|volume=220|issue=571–581|pages=291–333|doi=10.1098/rsta.1920.0009|bibcode=1920RSPTA.220..291D|issn=0264-3952|doi-access=free
}
}
<
/ref
>
<
ref
>
{
{
Cite
journal|last=Stanley|first=Matthew|date=2003-03-01|title='An
Expedition
to
Heal
the
Wounds
of
War
'
The
1919
Eclipse
and
Eddington
as
Quaker
Adventurer|url=https
:
//www.journals.uchicago.edu/doi/10.1086/376099|journal=Isis|volume=94|issue=1|pages=57–89|doi=10.1086/376099|pmid=12725104|bibcode=2003Isis
...
94
...
57S|s2cid=25615643|issn=0021-1753|access-date=2020-10-22|archive-date=2020-08-05|archive-url=https
:
//web.archive.org/web/20200805053416/https
:
//www.journals.uchicago.edu/doi/10.1086/376099|url-status=live
}
}
<
/ref
>
During
the
[
[
solar
eclipse
]
]
,
the
English
[
[
astronomer
]
]
and
physicist
[
[
Arthur
Eddington
]
]
observed
that
the
light
from
stars
passing
close
to
the
Sun
was
bent
.
The
effect
is
due
to
the
gravitational
attraction
of
light
by
the
Sun
.
The
observation
confirmed
that
the
energy
carried
by
light
indeed
is
equivalent
to
a
gravitational
mass
.
Another
seminal
experiment
,
the
[
[
Pound–Rebka
experiment
]
]
,
was
performed
in
1960.
<
ref
>
{
{
Cite
journal|last1=Pound|first1=R
.
V.|last2=Rebka|first2=G
.
A.|date=1960-04-01|title=Apparent
Weight
of
Photons|journal=Physical
Review
Letters|language=en|volume=4|issue=7|pages=337–341|doi=10.1103/PhysRevLett.4.337|bibcode=1960PhRvL
...
4..337P|issn=0031-9007|doi-access=free
}
}
<
/ref
>
In
this
test
a
beam
of
light
was
emitted
from
the
top
of
a
tower
and
detected
at
the
bottom
.
The
[
[
frequency
]
]
of
the
light
detected
was
higher
than
the
light
emitted
.
This
result
confirms
that
the
energy
of
photons
increases
when
they
fall
in
the
gravitational
field
of
the
Earth
.
The
energy
,
and
therefore
the
gravitational
mass
,
of
photons
is
proportional
to
their
frequency
as
stated
by
the
Planck
's
relation
.
==Efficiency==
In
some
reactions
,
matter
particles
can
be
destroyed
and
their
associated
energy
released
to
the
environment
as
other
forms
of
energy
,
such
as
light
and
heat.
<
ref
name=Serway1217
/
>
One
example
of
such
a
conversion
takes
place
in
elementary
particle
interactions
,
where
the
rest
energy
is
transformed
into
kinetic
energy.
<
ref
name=Serway1217
/
>
Such
conversions
between
types
of
energy
happen
in
nuclear
weapons
,
in
which
the
protons
and
neutrons
in
[
[
atomic
nuclei
]
]
lose
a
small
fraction
of
their
original
mass
,
though
the
mass
lost
is
not
due
to
the
destruction
of
any
smaller
constituents
.
[
[
Nuclear
fission
]
]
allows
a
tiny
fraction
of
the
energy
associated
with
the
mass
to
be
converted
into
usable
energy
such
as
radiation
;
in
the
decay
of
the
[
[
uranium
]
]
,
for
instance
,
about
0.1
%
of
the
mass
of
the
original
atom
is
lost.
<
ref
name=
''
bulletin1950
''
>
{
{
Cite
journal|last=Bethe|first=Hans
A.|date=1950-04-01|title=The
Hydrogen
Bomb|url=https
:
//doi.org/10.1080/00963402.1950.11461231|journal=Bulletin
of
the
Atomic
Scientists|volume=6|issue=4|pages=99–104|doi=10.1080/00963402.1950.11461231|bibcode=1950BuAtS
...
6d..99B|issn=0096-3402
}
}
<
/ref
>
In
theory
,
it
should
be
possible
to
destroy
matter
and
convert
all
of
the
rest-energy
associated
with
matter
into
heat
and
light
,
but
none
of
the
theoretically
known
methods
are
practical
.
One
way
to
harness
all
the
energy
associated
with
mass
is
to
annihilate
matter
with
[
[
antimatter
]
]
.
[
[
baryon
asymmetry|Antimatter
is
rare
in
our
universe
]
]
,
however
,
and
the
known
mechanisms
of
production
require
more
usable
energy
than
would
be
released
in
annihilation
.
[
[
CERN
]
]
estimated
in
2011
that
over
a
billion
times
more
energy
is
required
to
make
and
store
antimatter
than
could
be
released
in
its
annihilation.
<
ref
>
{
{
Cite
web|title=Making
antimatter
{
{
!
}
}
Angels
&
Demons
-
The
science
behind
the
story|url=https
:
//angelsanddemons.web.cern.ch/antimatter/making-antimatter.html|access-date=2020-10-15|website=angelsanddemons.web.cern.ch|archive-date=2020-11-01|archive-url=https
:
//web.archive.org/web/20201101023510/https
:
//angelsanddemons.web.cern.ch/antimatter/making-antimatter.html|url-status=live
}
}
<
/ref
>
As
most
of
the
mass
which
comprises
ordinary
objects
resides
in
protons
and
neutrons
,
converting
all
the
energy
of
ordinary
matter
into
more
useful
forms
requires
that
the
protons
and
neutrons
be
converted
to
lighter
particles
,
or
particles
with
no
mass
at
all
.
In
the
[
[
Standard
Model
of
particle
physics
]
]
,
the
[
[
baryon
number|number
of
protons
plus
neutrons
]
]
is
nearly
exactly
conserved
.
Despite
this
,
[
[
Gerard
't
Hooft
]
]
showed
that
there
is
a
process
that
converts
protons
and
neutrons
to
[
[
antielectron
]
]
s
and
[
[
neutrino
]
]
s.
<
ref
>
{
{
Cite
journal|last='t
Hooft|first=G.|date=1976-12-15|title=Computation
of
the
quantum
effects
due
to
a
four-dimensional
pseudoparticle|url=http
:
//dx.doi.org/10.1103/physrevd.14.3432|journal=Physical
Review
D|volume=14|issue=12|pages=3432–3450|doi=10.1103/physrevd.14.3432|bibcode=1976PhRvD..14.3432T|issn=0556-2821|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080229/https
:
//journals.aps.org/prd/abstract/10.1103/PhysRevD.14.3432|url-status=live
}
}
<
/ref
>
This
is
the
weak
[
[
SU
(
2
)
]
]
[
[
instanton
]
]
proposed
by
the
physicists
[
[
Alexander
Belavin
]
]
,
[
[
Alexander
Markovich
Polyakov
]
]
,
[
[
Albert
Schwarz
]
]
,
and
Yu
.
S.
Tyupkin.
<
ref
>
{
{
Cite
journal|last1=Belavin|first1=A.A.|last2=Polyakov|first2=A.M.|last3=Schwartz|first3=A.S.|last4=Tyupkin|first4=Yu.S.|date=October
1975|title=Pseudoparticle
solutions
of
the
Yang-Mills
equations|url=http
:
//dx.doi.org/10.1016/0370-2693
(
75
)
90163-x|journal=Physics
Letters
B|volume=59|issue=1|pages=85–87|doi=10.1016/0370-2693
(
75
)
90163-x|bibcode=1975PhLB
...
59
...
85B|issn=0370-2693|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080231/https
:
//www.sciencedirect.com/science/article/abs/pii/037026937590163X
?
via
%
3Dihub|url-status=live
}
}
<
/ref
>
This
process
,
can
in
principle
destroy
matter
and
convert
all
the
energy
of
matter
into
neutrinos
and
usable
energy
,
but
it
is
normally
extraordinarily
slow
.
It
was
later
shown
that
the
process
occurs
rapidly
at
extremely
high
temperatures
that
would
only
have
been
reached
shortly
after
the
[
[
Big
Bang
]
]
.
<
ref
>
{
{
cite
journal
|
last1
=
Klinkhammer
|
first1
=
F.
|
author-link2
=
Nicholas
Manton
|
last2
=
Manton
|
first2
=
N.
|
year
=
1984|
title
=
A
Saddle
Point
Solution
in
the
Weinberg
Salam
Theory
|
journal
=
Physical
Review
D
|
volume
=
30
|
issue
=
10|
page
=
2212
|
doi
=
10.1103/PhysRevD.30.2212
|
bibcode
=
1984PhRvD..30.2212K
}
}
<
/ref
>
Many
extensions
of
the
standard
model
contain
[
[
magnetic
monopole
]
]
s
,
and
in
some
models
of
[
[
grand
unification
theory|grand
unification
]
]
,
these
monopoles
catalyze
[
[
proton
decay
]
]
,
a
process
known
as
the
[
[
Callan–Rubakov
effect
]
]
.
<
ref
>
{
{
cite
journal
|
last1
=
Rubakov
|
first1
=
V.
A
.
|
year
=
1988
|
title
=
Monopole
Catalysis
of
Proton
Decay
|
journal
=
Reports
on
Progress
in
Physics
|
volume
=
51
|
issue
=
2|
pages
=
189–241
|
doi
=
10.1088/0034-4885/51/2/002
|
s2cid
=
250904729
}
}
<
/ref
>
This
process
would
be
an
efficient
mass–energy
conversion
at
ordinary
temperatures
,
but
it
requires
making
monopoles
and
anti-monopoles
,
whose
production
is
expected
to
be
inefficient
.
Another
method
of
completely
annihilating
matter
uses
the
gravitational
field
of
black
holes
.
The
British
[
[
theoretical
physicist
]
]
[
[
Stephen
Hawking
]
]
theorized
<
ref
>
{
{
cite
journal
|
last1
=
Hawking
|
first1
=
S.W
.
|
year
=
1974
|
title
=
Black
Holes
Explosions
?
|
journal
=
Nature
|
volume
=
248
|
issue
=
5443|
page
=
30
|
doi
=
10.1038/248030a0
|
bibcode
=
1974Natur.248
...
30H
|
s2cid
=
4290107
}
}
<
/ref
>
it
is
possible
to
throw
matter
into
a
black
hole
and
use
the
emitted
heat
to
generate
power
.
According
to
the
theory
of
[
[
Hawking
radiation
]
]
,
however
,
larger
black
holes
radiate
less
than
smaller
ones
,
so
that
usable
power
can
only
be
produced
by
small
black
holes
.
==Extension
for
systems
in
motion==
{
{
main|Energy–momentum
relation
}
}
Unlike
a
system
's
energy
in
an
inertial
frame
,
the
relativistic
energy
(
$
E
=
m
c
^2
$
)
of
a
system
depends
on
both
the
rest
mass
(
$
E
_{\
r
m
rel}
$
)
and
the
total
momentum
of
the
system
.
The
extension
of
Einstein
's
equation
to
these
systems
is
given
by
:
<
ref
>
{
{
Cite
book|last=Forshaw|first=Jeffrey
Robert|url=https
:
//www.worldcat.org/oclc/291193458|title=Dynamics
and
relativity|date=2009|publisher=John
Wiley
&
Sons|others=Smith
,
A.
Gavin.|isbn=978-0-470-01459-2|location=Chichester
,
UK|oclc=291193458|page=259|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080230/https
:
//www.worldcat.org/title/dynamics-and-relativity/oclc/291193458|url-status=live
}
}
<
/ref
>
<
ref
>
{
{
Cite
book|last=McMahon|first=David|url=https
:
//www.worldcat.org/oclc/61684277|title=Relativity
demystified|date=2006|publisher=McGraw-Hill|isbn=978-0-07-145545-9|location=New
York|oclc=61684277|chapter=1
:
Special
relativity
}
}
<
/ref
>
<
ref
group=
''
note
''
>
Some
authors
state
the
expression
equivalently
as
$
m
_0
$
where
$
E
= \
gamma
m
_0
c
^2
$
is
the
[
[
Lorentz
factor
]
]
.
<
/ref
>
:
$
\
gamma
$
or
$
E
_{\
r
m
rel}^2 - |\mathbf{
p
} |^2
c
^2 =
m
_0^2
c
^4 \\
E
_{\
r
m
rel}^2 - (
p
c
)^2 = (
m
_0
c
^2)^2
$
where
the
$
E
_{\
r
m
rel} = \sqrt{ (
m
_0
c
^2)^2 + (
p
c
)^2 } \,\!
$
term
represents
the
square
of
the
[
[
Euclidean
norm
]
]
(
total
vector
length
)
of
the
various
momentum
vectors
in
the
system
,
which
reduces
to
the
square
of
the
simple
momentum
magnitude
,
if
only
a
single
particle
is
considered
.
This
equation
is
called
the
[
[
energy–momentum
relation
]
]
and
reduces
to
$
(
p
c
)^2
$
when
the
momentum
term
is
zero
.
For
photons
where
$
E
_{\
r
m
rel} =
m
c
^2
$
,
the
equation
reduces
to
$
m
_0 = 0
$
.
==Low-speed
expansion==
Using
the
[
[
Lorentz
factor
]
]
,
{
{
math|
''
γ
''
}
}
,
the
energy–momentum
can
be
rewritten
as
{
{
math|
''
E
''
{
{
=
}
}
``
γmc
''
<
sup
>
2
<
/sup
>
}
}
and
expanded
as
a
[
[
power
series
]
]
:
:
$
E
_{\
r
m
rel} =
p
c
$
For
speeds
much
smaller
than
the
speed
of
light
,
higher-order
terms
in
this
expression
get
smaller
and
smaller
because
{
{
math|
{
{
sfrac|
''
v
''
|
''
c
''
}
}
}
}
is
small
.
For
low
speeds
,
all
but
the
first
two
terms
can
be
ignored
:
:
$
E
=
m
_0
c
^2 \left[1 + \frac{1}{2} \left(\frac{
v
}{
c
}\right)^2 + \frac{3}{8} \left(\frac{
v
}{
c
}\right)^4 + \frac{5}{16} \left(\frac{
v
}{
c
}\right)^6 + \ldots \right].
$
In
[
[
classical
mechanics
]
]
,
both
the
{
{
math|
''
m
''
<
sub
>
0
<
/sub
>
''
c
''
<
sup
>
2
<
/sup
>
}
}
term
and
the
high-speed
corrections
are
ignored
.
The
initial
value
of
the
energy
is
arbitrary
,
as
only
the
change
in
energy
can
be
measured
,
so
the
{
{
math|
''
m
''
<
sub
>
0
<
/sub
>
''
c
''
<
sup
>
2
<
/sup
>
}
}
term
is
ignored
in
classical
physics
.
While
the
higher-order
terms
become
important
at
higher
speeds
,
the
Newtonian
equation
is
a
highly
accurate
low-speed
approximation
;
adding
in
the
third
term
yields
:
:
$
E
\approx
m
_0
c
^2 + \frac{1}{2}
m
_0
v
^2.
$
.
The
difference
between
the
two
approximations
is
given
by
$
E
\approx
m
_0
c
^2 + \frac{1}{2}
m
_0
v
^2 \left(1 + \frac{3v^2}{4c^2}\right)
$
,
a
number
very
small
for
everyday
objects
.
In
2018
NASA
announced
the
[
[
Parker
Solar
Probe
]
]
was
the
fastest
ever
,
with
a
speed
of
{
{
convert|153,454|mph|m/s
}
}
.
<
ref
>
{
{
Cite
web|title=Parker
Solar
Probe
Becomes
Fastest-Ever
Spacecraft
–
Parker
Solar
Probe|url=https
:
//blogs.nasa.gov/parkersolarprobe/2018/10/29/parker-solar-probe-becomes-fastest-ever-spacecraft/|access-date=2020-08-25|website=blogs.nasa.gov|language=en-US|archive-date=2020-08-17|archive-url=https
:
//web.archive.org/web/20200817022956/https
:
//blogs.nasa.gov/parkersolarprobe/2018/10/29/parker-solar-probe-becomes-fastest-ever-spacecraft/|url-status=live
}
}
<
/ref
>
The
difference
between
the
approximations
for
the
Parker
Solar
Probe
in
2018
is
$
\tfrac{3v^2}{4c^2}
$
,
which
accounts
for
an
energy
correction
of
four
parts
per
hundred
million
.
The
[
[
gravitational
constant
]
]
,
in
contrast
,
has
a
standard
[
[
relative
uncertainty
]
]
of
about
$
\tfrac{3v^2}{4c^2} \approx 3.9 \times 10^{-8}
$
.
<
ref
>
{
{
Cite
web|title=CODATA
Value
:
Newtonian
constant
of
gravitation|url=https
:
//physics.nist.gov/cgi-bin/cuu/Value
?
bg|access-date=2020-08-25|website=physics.nist.gov|archive-date=2011-08-27|archive-url=https
:
//web.archive.org/web/20110827153649/http
:
//www.physics.nist.gov/cgi-bin/cuu/Value
?
bg|url-status=live
}
}
<
/ref
>
==Applications==
===Application
to
nuclear
physics===
{
{
main|Nuclear
binding
energy|Mass
defect
}
}
[
[
File
:
USS
Enterprise
(
CVAN-65
)
,
USS
Long
Beach
(
CGN-9
)
and
USS
Bainbridge
(
DLGN-25
)
underway
in
the
Mediterranean
Sea
during
Operation
Sea
Orbit
,
in
1964.jpg|thumb|right|Task
Force
One
,
the
world
's
first
nuclear-powered
task
force
.
{
{
USS|Enterprise|CVN-65|2
}
}
,
{
{
USS|Long
Beach|CGN-9|2
}
}
and
{
{
USS|Bainbridge|CGN-25|2
}
}
in
formation
in
the
Mediterranean
,
18
June
1964
.
''Enterprise
''
crew
members
are
spelling
out
Einstein
's
mass–energy
equivalence
formula
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
on
the
flight
deck
.
]
]
The
[
[
nuclear
binding
energy
]
]
is
the
minimum
energy
that
is
required
to
disassemble
the
nucleus
of
an
atom
into
its
component
parts.
<
ref
>
{
{
Cite
book|last=Rohlf|first=James
William.|url=https
:
//www.worldcat.org/oclc/29563946|title=Modern
physics
from
[
alpha
]
to
Z⁰|date=1994|publisher=John
Wiley|isbn=978-0-471-57270-1|edition=1st|location=New
York|oclc=29563946|page=20
}
}
<
/ref
>
The
mass
of
an
atom
is
less
than
the
sum
of
the
masses
of
its
constituents
due
to
the
attraction
of
the
[
[
strong
nuclear
force
]
]
.
<
ref
name=radiopharm
>
{
{
Citation|last=Rösch|first=Frank|title=The
Basics
of
Nuclear
Chemistry
and
Radiochemistry
:
An
Introduction
to
Nuclear
Transformations
and
Radioactive
Emissions|date=2019|url=https
:
//doi.org/10.1007/978-3-319-98947-1_3|work=Radiopharmaceutical
Chemistry|pages=27–61|editor-last=Lewis|editor-first=Jason
S.|place=Cham|publisher=Springer
International
Publishing|language=en|doi=10.1007/978-3-319-98947-1_3|isbn=978-3-319-98947-1|s2cid=134052082|access-date=2020-10-14|editor2-last=Windhorst|editor2-first=Albert
D.|editor3-last=Zeglis|editor3-first=Brian
M.
}
}
<
/ref
>
The
difference
between
the
two
masses
is
called
the
``
mass
defect
''
and
is
related
to
the
binding
energy
through
Einstein
's
formula.
<
ref
name=radiopharm
/
>
<
ref
>
{
{
Cite
book|last=Serway
,
Raymond
A.|title=Physics
for
scientists
and
engineers
with
modern
physics.|others=Jewett
,
John
W.
,
Peroomian
,
Vahé.|date=5
March
2013|isbn=978-1-133-95405-7|edition=9th|location=Boston
,
MA|oclc=802321453|page=1419
}
}
<
/ref
>
<
ref
>
{
{
Cite
book|last1=Frisch|first1=David
H|url=https
:
//www.worldcat.org/oclc/222569|title=Elementary
particles|last2=Thorndike|first2=Alan
M|date=1964|publisher=D
.
Van
Nostrand|location=Princeton
,
N.J.|language=en|oclc=222569|pages=11–12
}
}
<
/ref
>
The
principle
is
used
in
modeling
nuclear
fission
reactions
and
it
implies
a
great
amount
of
energy
can
be
released
by
the
nuclear
fission
[
[
chain
reaction
]
]
s
used
in
both
[
[
nuclear
weapon
]
]
s
and
[
[
nuclear
power
]
]
.
A
water
molecule
weighs
a
little
less
than
two
free
hydrogen
atoms
and
an
oxygen
atom
.
The
minuscule
mass
difference
is
the
energy
needed
to
split
the
molecule
into
three
individual
atoms
(
divided
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
)
,
which
was
given
off
as
heat
when
the
molecule
formed
(
this
heat
had
mass
)
.
Similarly
,
a
stick
of
dynamite
in
theory
weighs
a
little
bit
more
than
the
fragments
after
the
explosion
;
in
this
case
the
mass
difference
is
the
energy
and
heat
that
is
released
when
the
dynamite
explodes
.
Such
a
change
in
mass
may
only
happen
when
the
system
is
open
,
and
the
energy
and
mass
are
allowed
to
escape
.
Thus
,
if
a
stick
of
dynamite
is
blown
up
in
a
hermetically
sealed
chamber
,
the
mass
of
the
chamber
and
fragments
,
the
heat
,
sound
,
and
light
would
still
be
equal
to
the
original
mass
of
the
chamber
and
dynamite
.
If
sitting
on
a
scale
,
the
weight
and
mass
would
not
change
.
This
would
in
theory
also
happen
even
with
a
nuclear
bomb
,
if
it
could
be
kept
in
an
ideal
box
of
infinite
strength
,
which
did
not
rupture
or
pass
[
[
radiation
]
]
.
{
{
refn|group=note|name=
''
A
.
Wheeler
,
1992.
pp
.
248
''
|See
Taylor
and
Wheeler
<
ref
>
{
{
Cite
book|last=Taylor
,
Edwin
F.|url=https
:
//www.worldcat.org/oclc/25165077|title=Spacetime
physics
:
introduction
to
special
relativity
|date=1992|publisher=W.H
.
Freeman|others=Wheeler
,
John
Archibald
,
1911-2008.|isbn=978-0-7167-2327-1|edition=2nd|location=New
York|oclc=25165077|pages=248–249
}
}
<
/ref
>
for
a
discussion
of
mass
remaining
constant
after
detonation
of
nuclear
bombs
,
until
heat
is
allowed
to
escape
.
}
}
Thus
,
a
21.5
&
nbsp
;
[
[
TNT
equivalent|kiloton
]
]
(
{
{
val|9|e=13|u=joule
}
}
)
nuclear
bomb
produces
about
one
gram
of
heat
and
electromagnetic
radiation
,
but
the
mass
of
this
energy
would
not
be
detectable
in
an
exploded
bomb
in
an
ideal
box
sitting
on
a
scale
;
instead
,
the
contents
of
the
box
would
be
heated
to
millions
of
degrees
without
changing
total
mass
and
weight
.
If
a
transparent
window
passing
only
electromagnetic
radiation
were
opened
in
such
an
ideal
box
after
the
explosion
,
and
a
beam
of
X-rays
and
other
lower-energy
light
allowed
to
escape
the
box
,
it
would
eventually
be
found
to
weigh
one
gram
less
than
it
had
before
the
explosion
.
This
weight
loss
and
mass
loss
would
happen
as
the
box
was
cooled
by
this
process
,
to
room
temperature
.
However
,
any
surrounding
mass
that
absorbed
the
X-rays
(
and
other
``
heat
''
)
would
``
gain
''
this
gram
of
mass
from
the
resulting
heating
,
thus
,
in
this
case
,
the
mass
``
loss
''
would
represent
merely
its
relocation
.
===Practical
examples===
Einstein
used
the
[
[
centimetre–gram–second
system
of
units
]
]
(
cgs
)
,
but
the
formula
is
independent
of
the
system
of
units
.
In
natural
units
,
the
numerical
value
of
the
speed
of
light
is
set
to
equal
1
,
and
the
formula
expresses
an
equality
of
numerical
values
:
{
{
math|1=
''
E
''
=
``
m
''
}
}
.
In
the
[
[
International
System
of
Units|SI
]
]
system
(
expressing
the
ratio
{
{
math|
{
{
sfrac|
''
E
''
|
''
m
''
}
}
}
}
in
[
[
joules
]
]
per
kilogram
using
the
value
of
{
{
math|
''
c
''
}
}
in
[
[
metre
per
second|meters
per
second
]
]
)
:
<
ref
>
{
{
cite
book
|title=Megawatts
and
Megatons
:
The
Future
of
Nuclear
Power
and
Nuclear
Weapons
|edition=illustrated
|first1=Richard
L.
|last1=Garwin
|first2=Georges
|last2=Charpak
|publisher=University
of
Chicago
Press
|year=2002
|isbn=978-0-226-28427-9
|page=
[
https
:
//books.google.com/books
?
id=1YgBR6shTckC
&
pg=PA17
17
]
|url=https
:
//books.google.com/books
?
id=1YgBR6shTckC
}
}
<
/ref
>
:
{
{
math|1=
{
{
Sfrac|
''
E
''
|
''
m
''
}
}
=
}
}
{
{
math|1=
''
c
''
<
sup
>
2
<
/sup
>
=
(
{
{
val|299792458|u=m/s
}
}
)
<
sup
>
2
<
/sup
>
=
}
}
{
{
math|
{
{
val|89875517873681764|u=J/kg
}
}
}
}
(
≈
&
nbsp
;
9.0
×
10
<
sup
>
16
<
/sup
>
joules
per
kilogram
)
.
So
the
energy
equivalent
of
one
kilogram
of
mass
is
*89.9
&
nbsp
;
[
[
joules|petajoules
]
]
*25.0
&
nbsp
;
billion
[
[
kilowatt-hour
]
]
s
(
≈
&
nbsp
;
25,000
&
nbsp
;
[
[
GW·h
]
]
)
*21.5
&
nbsp
;
trillion
[
[
calorie|kilocalories
]
]
(
≈
&
nbsp
;
21
&
nbsp
;
Pcal
)
<
ref
group=
''
note
''
name=
''
Conversion
''
>
Conversions
used
:
1956
International
(
Steam
)
Table
(
IT
)
values
where
one
calorie
≡
&
nbsp
;
4.1868
&
nbsp
;
J
and
one
BTU
≡
&
nbsp
;
1055.05585262
&
nbsp
;
J
.
Weapons
designers
'
conversion
value
of
one
&
nbsp
;
gram
TNT
≡
1000
&
nbsp
;
calories
used.
<
/ref
>
*85.2
&
nbsp
;
trillion
[
[
British
thermal
unit|BTUs
]
]
<
ref
group=
''
note
''
name=
''
Conversion
''
/
>
*0.0852
[
[
Quad
(
unit
)
|quads
]
]
or
the
energy
released
by
combustion
of
the
following
:
*21
500
&
nbsp
;
[
[
kiloton
]
]
s
of
[
[
TNT
equivalent|TNT-equivalent
]
]
energy
(
≈
&
nbsp
;
21
&
nbsp
;
Mt
)
<
ref
group=
''
note
''
name=
''
Conversion
''
/
>
*
{
{
val|2630000000
}
}
[
[
litre
]
]
s
or
{
{
val|695000000
}
}
US
[
[
gallon
]
]
s
of
automotive
[
[
Gasoline
#
Combustion
energy
content|gasoline
]
]
Any
time
energy
is
released
,
the
process
can
be
evaluated
from
an
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
perspective
.
For
instance
,
the
``
[
[
Fat
Man|Gadget
]
]
''
-style
bomb
used
in
the
[
[
Trinity
test
]
]
and
the
[
[
bombing
of
Nagasaki
]
]
had
an
explosive
yield
equivalent
to
21
&
nbsp
;
kt
of
TNT.
<
ref
>
{
{
Cite
web|last=John|first=Malik|date=September
1985|title=The
Yields
of
the
Hiroshima
and
Nagasaki
Nuclear
Explosions|url=https
:
//permalink.lanl.gov/object/tr
?
what=info
:
lanl-repo/lareport/LA-08819|access-date=1
October
2020|website=Los
Alamos
National
Laboratories|archive-date=13
October
2020|archive-url=https
:
//web.archive.org/web/20201013123611/https
:
//permalink.lanl.gov/object/tr
?
what=info
%
3Alanl-repo
%
2Flareport
%
2FLA-08819|url-status=live
}
}
<
/ref
>
About
1
&
nbsp
;
kg
of
the
approximately
6.15
&
nbsp
;
kg
of
[
[
plutonium
]
]
in
each
of
these
bombs
fissioned
into
lighter
elements
totaling
almost
exactly
one
gram
less
,
after
cooling
.
The
electromagnetic
radiation
and
kinetic
energy
(
thermal
and
blast
energy
)
released
in
this
explosion
carried
the
missing
gram
of
mass
.
Whenever
energy
is
added
to
a
system
,
the
system
gains
mass
,
as
shown
when
the
equation
is
rearranged
:
*
A
[
[
spring
(
device
)
|spring
's
]
]
mass
increases
whenever
it
is
put
into
compression
or
tension
.
Its
mass
increase
arises
from
the
increased
potential
energy
stored
within
it
,
which
is
bound
in
the
stretched
chemical
(
electron
)
bonds
linking
the
atoms
within
the
spring
.
*
Raising
the
temperature
of
an
object
(
increasing
its
[
[
thermal
energy
]
]
)
increases
its
mass
.
For
example
,
consider
the
world
's
primary
mass
standard
for
the
kilogram
,
made
of
[
[
platinum
]
]
and
[
[
iridium
]
]
.
If
its
temperature
is
allowed
to
change
by
1
&
nbsp
;
°C
,
its
mass
changes
by
1.5
picograms
(
1
&
nbsp
;
pg
=
{
{
val|1|e=-12|u=g
}
}
)
.
<
ref
group=note
>
Assuming
a
90/10
alloy
of
Pt/Ir
by
weight
,
a
{
{
math|
''
C
<
sub
>
p
<
/sub
>
''
}
}
of
25.9
for
Pt
and
25.1
for
Ir
,
a
Pt-dominated
average
{
{
math|
''
C
<
sub
>
p
<
/sub
>
''
}
}
of
25.8
,
5.134
&
nbsp
;
moles
of
metal
,
and
132
&
nbsp
;
J⋅K
<
sup
>
−1
<
/sup
>
for
the
prototype
.
A
variation
of
±1.5
&
nbsp
;
picograms
is
much
smaller
than
the
uncertainty
in
the
mass
of
the
international
prototype
,
which
is
±2
&
nbsp
;
micrograms.
<
/ref
>
*
A
spinning
ball
has
greater
mass
than
when
it
is
not
spinning
.
Its
increase
of
mass
is
exactly
the
equivalent
of
the
mass
of
[
[
Rotational
energy|energy
of
rotation
]
]
,
which
is
itself
the
sum
of
the
kinetic
energies
of
all
the
moving
parts
of
the
ball
.
For
example
,
[
[
the
Earth
]
]
itself
is
more
massive
due
to
its
rotation
,
than
it
would
be
with
no
rotation
.
The
rotational
energy
of
the
Earth
is
greater
than
10
<
sup
>
24
<
/sup
>
Joules
,
which
is
over
10
<
sup
>
7
<
/sup
>
kg.
<
ref
>
{
{
Cite
news|last=Allain|first=Rhett|date=2009-06-22|title=Rotational
Energy
of
the
Earth
as
an
energy
source|magazine=Wired|url=https
:
//www.wired.com/2009/06/rotational-energy-of-the-earth-as-an-energy-source/|access-date=2020-10-14|issn=1059-1028|archive-date=2020-10-16|archive-url=https
:
//web.archive.org/web/20201016032556/https
:
//www.wired.com/2009/06/rotational-energy-of-the-earth-as-an-energy-source/|url-status=live
}
}
<
/ref
>
==History==
{
{
further|History
of
special
relativity
}
}
While
Einstein
was
the
first
to
have
correctly
deduced
the
mass–energy
equivalence
formula
,
he
was
not
the
first
to
have
related
energy
with
mass
,
though
nearly
all
previous
authors
thought
that
the
energy
that
contributes
to
mass
comes
only
from
electromagnetic
fields.
<
ref
>
{
{
Cite
book|last=Whittaker
,
E.
T.|author-link=E
.
T.
Whittaker|title=
[
[
A
History
of
the
Theories
of
Aether
and
Electricity
]
]
|date=1989|publisher=Dover
Publications|isbn=978-0-486-26126-3|location=New
York|oclc=20357018|volume=2|chapter=The
relativity
theory
of
Poincaré
and
Lorentz
}
}
<
/ref
>
<
ref
name=
''
mill
''
>
{
{
Cite
book|last=Miller
,
Arthur
I.|url=https
:
//www.worldcat.org/oclc/5894058|title=Albert
Einstein
's
special
theory
of
relativity
:
emergence
(
1905
)
and
early
interpretation
,
1905-1911|date=1981|publisher=Addison-Wesley
Pub
.
Co.
,
Advanced
Book
Program|isbn=978-0-201-04680-9|location=Reading
,
Mass.|section=Some
Others
Who
Discussed
an
Association
Between
Energy
and
Mass|oclc=5894058|pages=339–340
}
}
<
/ref
>
<
ref
name=
''
darr
''
>
{
{
Citation|author=Darrigol
,
O.|chapter-url=https
:
//www.worldcat.org/oclc/317084635|title=Einstein
,
1905-2005
:
Poincaré
Seminar
2005|chapter=The
Genesis
of
the
theory
of
relativity|date=2006|publisher=Birkhäuser
Verlag|others=Damour
,
Thibault.|isbn=978-3-7643-7436-5|location=Basel|oclc=317084635|pages=1–22|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080237/https
:
//www.worldcat.org/title/einstein-1905-2005-poincare-seminar-2005/oclc/317084635|url-status=live
}
}
<
/ref
>
Once
discovered
,
Einstein
's
formula
was
initially
written
in
many
different
notations
,
and
its
interpretation
and
justification
was
further
developed
in
several
steps.
<
ref
name=jammer2
/
>
<
ref
name=hecht
>
{
{
Cite
journal|last=Hecht|first=Eugene|date=June
2011|title=How
Einstein
confirmed
E0=mc2|url=http
:
//aapt.scitation.org/doi/10.1119/1.3549223|journal=American
Journal
of
Physics|language=en|volume=79|issue=6|pages=591–600|doi=10.1119/1.3549223|bibcode=2011AmJPh..79..591H|issn=0002-9505|access-date=2020-10-14|archive-date=2019-04-05|archive-url=https
:
//web.archive.org/web/20190405015020/https
:
//aapt.scitation.org/doi/10.1119/1.3549223|url-status=live
}
}
<
/ref
>
===Developments
prior
to
Einstein===
[
[
File
:
Portrait
of
Sir
Isaac
Newton
,
1689.jpg|thumb|In
the
revised
English
edition
of
[
[
Isaac
Newton
]
]
's
``
[
[
Opticks
]
]
''
,
published
in
1717
,
Newton
speculated
on
the
equivalence
of
mass
and
light
.
]
]
Eighteenth
century
theories
on
the
correlation
of
mass
and
energy
included
that
devised
by
the
English
scientist
[
[
Isaac
Newton
]
]
in
1717
,
who
speculated
that
light
particles
and
matter
particles
were
interconvertible
in
``
Query
30
''
of
the
``
[
[
Opticks
]
]
''
,
where
he
asks
:
``
Are
not
the
gross
bodies
and
light
convertible
into
one
another
,
and
may
not
bodies
receive
much
of
their
activity
from
the
particles
of
light
which
enter
their
composition
?
``
<
ref
>
{
{
Cite
web|title=Selected
Queries
from
Isaac
Newton
's
Opticks
{
{
!
}
}
Inters.org|url=http
:
//inters.org/newton-opticks-queries|access-date=2020-10-14|website=inters.org|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080253/http
:
//inters.org/newton-opticks-queries|url-status=live
}
}
<
/ref
>
Swedish
scientist
and
theologian
[
[
Emanuel
Swedenborg
]
]
,
in
his
``
[
[
The
Principia
(
book
)
|Principia
]
]
''
of
1734
theorized
that
all
matter
is
ultimately
composed
of
dimensionless
points
of
``
pure
and
total
motion
''
.
He
described
this
motion
as
being
without
force
,
direction
or
speed
,
but
having
the
potential
for
force
,
direction
and
speed
everywhere
within
it.
<
ref
>
{
{
Cite
book|last=Swedenborg|first=Emanuel|url=https
:
//books.google.com/books
?
id=1keP6ZXitBYC|title=Principia
rerum
naturalium
sive
Novorum
tentaminum
phaenomena
mundi
elementaris
philosophice
explicandi
...
|date=1734|publisher=sumptibus
Friderici
Hekelii|language=la|page=32|chapter=De
Simplici
Mundi
vel
Puncto
naturali
}
}
<
/ref
>
<
ref
>
{
{
Cite
book|last=Swedenborg|first=Emanuel|url=https
:
//www.worldcat.org/oclc/863755|title=The
principia
:
or
,
The
first
principles
of
natural
things
,
being
new
attempts
toward
a
philosophical
explanation
of
the
elementary
world.|translator-last=Clissold|translator-first=Augustus|translator-link=Augustus
Clissold|date=1845|pages=55–57|publisher=W
.
Newbery
;
O.
Clapp|location=London
;
Boston|language=en|oclc=863755|access-date=2020-10-14
}
}
<
/ref
>
During
the
nineteenth
century
there
were
several
speculative
attempts
to
show
that
mass
and
energy
were
proportional
in
various
[
[
Aether
theories|ether
theories
]
]
.
<
ref
>
{
{
Cite
book|last=Kragh|first=Helge|url=https
:
//www.jstor.org/stable/j.ctv10crfmk|title=Quantum
generations
:
a
history
of
physics
in
the
twentieth
century|pages=3–12|date=1999|isbn=978-0-691-21419-1|language=en|oclc=1159003206|chapter=Fin-de-Siècle
Physics
:
A
World
Picture
in
Flux|publisher=Princeton
University
Press|doi=10.2307/j.ctv10crfmk
|jstor=j.ctv10crfmk|s2cid=243126061
}
}
<
/ref
>
In
1873
the
Russian
physicist
and
mathematician
[
[
Nikolay
Umov
]
]
pointed
out
a
relation
between
mass
and
energy
for
ether
in
the
form
of
{
{
math|1=
''
Е
''
=
``
kmc
''
<
sup
>
2
<
/sup
>
}
}
,
where
{
{
math|0.5
≤
``
k
''
≤
1
}
}
.
<
ref
>
''Умов
Н
.
А
.
''
Избранные
сочинения
[
N.A
.
Umov
.
Selected
Works
]
.
(
1950
)
М
.
—
Л..
(
in
Russian
)
<
/ref
>
The
writings
of
the
English
engineer
[
[
Samuel
Tolver
Preston
]
]
,
<
ref
>
{
{
Cite
book|last=Preston|first=S
.
Tolver|url=https
:
//www.worldcat.org/oclc/5834362|title=Physics
of
the
ether|date=1875|publisher=E
.
&
F.N
.
Spon|location=London
;
New
York|language=en|oclc=5834362|access-date=23
October
2020
}
}
<
/ref
>
and
a
1903
paper
by
the
Italian
industrialist
and
[
[
geologist
]
]
[
[
Olinto
De
Pretto
]
]
,
<
ref
>
{
{
Cite
book|last1=Bartocci|first1=U|url=https
:
//www.worldcat.org/oclc/44897464|title=Albert
Einstein
e
Olinto
De
Pretto
:
la
vera
storia
della
formula
più
famosa
del
mondo|last2=Bonicelli|first2=Bianca
Maria|date=1999|publisher=Andromeda|location=Bologna|language=it|oclc=44897464|access-date=2020-10-14
}
}
<
/ref
>
<
ref
>
{
{
Cite
news|last=Carroll|first=Rory|date=1999-11-11|title=Einstein
's
E=mc2
'was
Italian
's
idea'|language=en-GB|work=The
Guardian|url=https
:
//www.theguardian.com/world/1999/nov/11/rorycarroll|access-date=2020-10-23|issn=0261-3077|archive-date=2020-10-23|archive-url=https
:
//web.archive.org/web/20201023034128/https
:
//www.theguardian.com/world/1999/nov/11/rorycarroll|url-status=live
}
}
<
/ref
>
presented
a
mass–energy
relation
.
Italian
mathematician
and
math
historian
[
[
Umberto
Bartocci
]
]
observed
that
there
were
only
[
[
six
degrees
of
separation|three
degrees
of
separation
]
]
linking
De
Pretto
to
Einstein
,
concluding
that
Einstein
was
probably
aware
of
De
Pretto
's
work.
<
ref
>
{
{
Cite
book|last1=Bartocci|first1=U|url=https
:
//www.worldcat.org/oclc/44897464|title=Albert
Einstein
e
Olinto
De
Pretto
:
la
vera
storia
della
formula
più
famosa
del
mondo|last2=Bonicelli|first2=Bianca
Maria|date=1999|publisher=Andromeda|location=Bologna|language=it|oclc=44897464|section=Pretto
,
O
.
''Reale
Instituto
Veneto
Di
Scienze
,
Lettere
Ed
Arti
''
,
LXIII
,
II
,
439–500|access-date=2020-10-14
}
}
<
/ref
>
<
ref
>
{
{
Cite
web|url=http
:
//www.cartesio-episteme.net/fis/depret-bombay.htm|title=Information
about
the
``
De
Pretto-Einstein
case
''
|website=www.cartesio-episteme.net
}
}
<
/ref
>
Preston
and
De
Pretto
,
following
physicist
[
[
Georges-Louis
Le
Sage
]
]
,
imagined
that
the
universe
was
filled
with
an
[
[
Aether
(
classical
element
)
|ether
]
]
of
tiny
particles
that
always
move
at
speed
{
{
mvar|c
}
}
.
Each
of
these
particles
has
a
kinetic
energy
of
{
{
math|
''
mc
''
<
sup
>
2
<
/sup
>
}
}
up
to
a
small
numerical
factor
.
The
nonrelativistic
kinetic
energy
formula
did
not
always
include
the
traditional
factor
of
{
{
sfrac|2
}
}
,
since
German
polymath
[
[
Gottfried
Leibniz
]
]
introduced
kinetic
energy
without
it
,
and
the
{
{
sfrac|2
}
}
is
largely
conventional
in
prerelativistic
physics.
<
ref
>
{
{
Cite
journal|last=Prentis|first=Jeffrey
J.|date=August
2005|title=Why
is
the
energy
of
motion
proportional
to
the
square
of
the
velocity
?
|url=http
:
//aapt.scitation.org/doi/10.1119/1.1927550|journal=American
Journal
of
Physics|language=en|volume=73|issue=8|pages=701–707|doi=10.1119/1.1927550|bibcode=2005AmJPh..73..701P|issn=0002-9505
}
}
<
/ref
>
By
assuming
that
every
particle
has
a
mass
that
is
the
sum
of
the
masses
of
the
ether
particles
,
the
authors
concluded
that
all
matter
contains
an
amount
of
kinetic
energy
either
given
by
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
or
{
{
math|1=2
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
depending
on
the
convention
.
A
particle
ether
was
usually
considered
unacceptably
speculative
science
at
the
time
,
<
ref
>
{
{
Cite
journal|last=Worrall|first=John|date=1985-03-01|title=Reviews|url=https
:
//academic.oup.com/bjps/article/36/1/81/1477386|journal=The
British
Journal
for
the
Philosophy
of
Science|language=en|volume=36|issue=1|pages=81–85|doi=10.1093/bjps/36.1.81|issn=0007-0882|access-date=2020-10-14
}
}
<
/ref
>
and
since
these
authors
did
not
formulate
relativity
,
their
reasoning
is
completely
different
from
that
of
Einstein
,
who
used
relativity
to
change
frames
.
In
1905
,
and
independent
of
Einstein
,
French
polymath
[
[
Gustave
Le
Bon
]
]
speculated
that
atoms
could
release
large
amounts
of
latent
energy
,
reasoning
from
an
all-encompassing
qualitative
philosophy
of
physics.
<
ref
>
{
{
Cite
book|last=Le
Bon|first=Gustave|url=https
:
//www.worldcat.org/oclc/875679536|title=The
evolution
of
forces|at=
[
http
:
//www.rexresearch.com/lebonfor/evforp1.htm
#
p1b3ch2
The
Energetical
Explanation
of
Phenomena
]
|date=2014|isbn=978-1-4942-9965-1|language=en|oclc=875679536|access-date=2020-10-14
}
}
<
/ref
>
<
ref
>
{
{
Cite
journal|last=Bizouard|first=Christian|date=2004|title=E
=
mc2
l'équation
de
Poincaré
,
Einstein
et
Planck
:
Henri
Poincare
et
la
physique|url=https
:
//pascal-francis.inist.fr/vibad/index.php
?
action=getRecordDetail
&
idt=16384743|journal=E
=
mc2
l'équation
de
Poincaré
,
Einstein
et
Planck
:
Henri
Poincare
et
la
physique|issue=4|pages=35–37|issn=0151-0304
}
}
<
/ref
>
====Electromagnetic
mass====
{
{
Main|Electromagnetic
mass
}
}
There
were
many
attempts
in
the
19th
and
the
beginning
of
the
20th
century—like
those
of
British
physicists
[
[
J.
J.
Thomson
]
]
in
1881
and
[
[
Oliver
Heaviside
]
]
in
1889
,
and
[
[
George
Frederick
Charles
Searle
]
]
in
1897
,
German
physicists
[
[
Wilhelm
Wien
]
]
in
1900
and
[
[
Max
Abraham
]
]
in
1902
,
and
the
Dutch
physicist
[
[
Hendrik
Antoon
Lorentz
]
]
in
1904—to
understand
how
the
mass
of
a
charged
object
depends
on
the
[
[
electrostatic
field
]
]
.
<
ref
>
{
{
Cite
book|last=Whittaker
,
E.
T.|author-link=E
.
T.
Whittaker|title=
[
[
A
History
of
the
Theories
of
Aether
and
Electricity
]
]
|date=1989|publisher=Dover
Publications|isbn=978-0-486-26126-3|location=New
York|oclc=20357018|volume=1|chapter=The
followeres
of
Maxwell
}
}
<
/ref
>
This
concept
was
called
[
[
electromagnetic
mass
]
]
,
and
was
considered
as
being
dependent
on
velocity
and
direction
as
well
.
Lorentz
in
1904
gave
the
following
expressions
for
longitudinal
and
transverse
electromagnetic
mass
:
:
$
2.2 \times 10^{-5}
$
,
where
:
$
m
_{
L
}=\frac{
m
_{0}}{\left(\sqrt{1-\frac{
v
^{2}}{
c
^{2}}}\right)^{3}},\quad
m
_{
T
}=\frac{
m
_{0}}{\sqrt{1-\frac{
v
^{2}}{
c
^{2}}}}
$
Another
way
of
deriving
a
type
of
electromagnetic
mass
was
based
on
the
concept
of
[
[
radiation
pressure
]
]
.
In
1900
,
French
polymath
[
[
Henri
Poincaré
]
]
associated
electromagnetic
radiation
energy
with
a
``
fictitious
fluid
''
having
momentum
and
mass
<
ref
name=action
/
>
:
$
m
_{0}=\frac{4}{3}\frac{
E
_{
e
m
}}{
c
^{2}}
$
By
that
,
Poincaré
tried
to
save
the
center
of
mass
theorem
in
Lorentz
's
theory
,
though
his
treatment
led
to
radiation
paradoxes.
<
ref
name=
''
darr
''
/
>
Austrian
physicist
[
[
Friedrich
Hasenöhrl
]
]
showed
in
1904
that
electromagnetic
[
[
cavity
radiation
]
]
contributes
the
``
apparent
mass
''
:
$
m
_{
e
m
}=\frac{
E
_{
e
m
}}{
c
^2}\,.
$
to
the
cavity
's
mass
.
He
argued
that
this
implies
mass
dependence
on
temperature
as
well.
<
ref
>
{
{
Cite
web|date=2011-08-23|title=Did
Einstein
discover
E
=
mc2
?
|url=https
:
//physicsworld.com/a/did-einstein-discover-e-mc2/|access-date=2020-10-14|website=
[
[
Physics
World
]
]
|language=en-GB|archive-date=2020-10-16|archive-url=https
:
//web.archive.org/web/20201016144605/https
:
//physicsworld.com/a/did-einstein-discover-e-mc2/|url-status=live
}
}
<
/ref
>
===Einstein
:
mass–energy
equivalence===
[
[
File
:
Einstein
1921
by
F
Schmutzer
-
restoration.jpg|thumb|Photo
of
[
[
Albert
Einstein
]
]
in
1921
]
]
Einstein
did
not
write
the
exact
formula
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
in
his
1905
[
[
Annus
Mirabilis
Papers|
''
Annus
Mirabilis
''
paper
]
]
``
Does
the
Inertia
of
an
object
Depend
Upon
Its
Energy
Content
?
``
;
<
ref
name=
''
inertia
''
/
>
rather
,
the
paper
states
that
if
a
body
gives
off
the
energy
{
{
mvar|L
}
}
in
the
form
of
radiation
,
its
mass
diminishes
by
{
{
math|
{
{
sfrac|
''
L
''
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
.
<
ref
group=note
>
Here
,
``
radiation
''
means
[
[
electromagnetic
radiation
]
]
,
or
light
,
and
mass
means
the
ordinary
Newtonian
mass
of
a
slow-moving
object.
<
/ref
>
This
formulation
relates
only
a
change
{
{
math|Δ
''
m
''
}
}
in
mass
to
a
change
{
{
mvar|L
}
}
in
energy
without
requiring
the
absolute
relationship
.
The
relationship
convinced
him
that
mass
and
energy
can
be
seen
as
two
names
for
the
same
underlying
,
conserved
physical
quantity.
<
ref
>
{
{
Cite
journal|last=Hecht|first=Eugene|date=September
2009|title=Einstein
on
mass
and
energy|url=http
:
//aapt.scitation.org/doi/10.1119/1.3160671|journal=American
Journal
of
Physics|language=en|volume=77|issue=9|pages=799–806|doi=10.1119/1.3160671|bibcode=2009AmJPh..77..799H|issn=0002-9505|quote=Einstein
was
unequivocally
against
the
traditional
idea
of
conservation
of
mass
.
He
had
concluded
that
mass
and
energy
were
essentially
one
and
the
same
;
'inert
mass
is
simply
latent
energy
.
'
He
made
his
position
known
publicly
time
and
again…|access-date=2020-10-14|archive-date=2019-05-28|archive-url=https
:
//web.archive.org/web/20190528171923/https
:
//aapt.scitation.org/doi/10.1119/1.3160671|url-status=live
}
}
<
/ref
>
He
has
stated
that
the
laws
of
conservation
of
energy
and
conservation
of
mass
are
``
one
and
the
same
''
.
<
ref
>
{
{
Cite
journal|last=Einstein|first=Albert|date=1940-05-24|title=Considerations
Concerning
the
Fundaments
of
Theoretical
Physics|url=https
:
//www.science.org/doi/10.1126/science.91.2369.487|journal=Science|language=en|volume=91|issue=2369|pages=487–492|doi=10.1126/science.91.2369.487|issn=0036-8075|pmid=17847438|bibcode=1940Sci
...
.91..487E|quote=There
followed
also
the
principle
of
the
equivalence
of
mass
and
energy
,
with
the
laws
of
conservation
of
mass
and
energy
becoming
one
and
the
same.|access-date=2020-10-14|archive-date=2020-07-11|archive-url=https
:
//web.archive.org/web/20200711070127/https
:
//science.sciencemag.org/content/91/2369/487|url-status=live
}
}
<
/ref
>
Einstein
elaborated
in
a
1946
essay
that
``
the
principle
of
the
conservation
of
mass…
proved
inadequate
in
the
face
of
the
special
theory
of
relativity
.
It
was
therefore
merged
with
the
energy
[
[
conservation
law|conservation
]
]
principle—just
as
,
about
60
years
before
,
the
principle
of
the
[
[
conservation
of
mechanical
energy
]
]
had
been
combined
with
the
principle
of
the
conservation
of
heat
[
thermal
energy
]
.
We
might
say
that
the
principle
of
the
conservation
of
energy
,
having
previously
swallowed
up
that
of
the
conservation
of
heat
,
now
proceeded
to
swallow
that
of
the
conservation
of
mass—and
holds
the
field
alone
.
``
<
ref
>
{
{
cite
book|url=
https
:
//books.google.com/books
?
id=SYPbH6xCbUMC
&
pg=PA14
|first=Albert
|last=Einstein
|title=The
Theory
of
Relativity
(
And
Other
Essays
)
|publisher=Citadel
Press
|year=1950
|page=14|isbn=978-0-8065-1765-0
}
}
<
/ref
>
====Mass–velocity
relationship====
[
[
File
:
E
mc
2
IMG
0859.jpg|thumb|The
equation
in
[
[
Albert
Einstein
]
]
's
own
handwriting
from
1912
]
]
In
developing
[
[
special
relativity
]
]
,
Einstein
found
that
the
[
[
kinetic
energy
#
Relativistic
kinetic
energy
of
rigid
bodies|kinetic
energy
]
]
of
a
moving
body
is
:
$
m
_{0}=\frac{4}{3}\frac{
E
_{
e
m
}}{
c
^{2}}
$
with
{
{
math|
''
v
''
}
}
the
[
[
velocity
]
]
,
{
{
math|
''
m
''
{
{
sub|0
}
}
}
}
the
rest
mass
,
and
{
{
math|
''
γ
''
}
}
the
Lorentz
factor
.
He
included
the
second
term
on
the
right
to
make
sure
that
for
small
velocities
the
energy
would
be
the
same
as
in
classical
mechanics
,
thus
satisfying
the
[
[
correspondence
principle
]
]
:
:
<
math
id=kineticEnergy
>
E_k
=
\frac
{
1
}
{
2
}
m_0
v^2
+
\cdots
<
/math
>
Without
this
second
term
,
there
would
be
an
additional
contribution
in
the
energy
when
the
particle
is
not
moving
.
====Einsteins
's
view
on
mass====
{
{
see
also|Mass
in
special
relativity
#
Early
developments
:
transverse
and
longitudinal
mass
}
}
Einstein
,
following
Lorentz
and
Abraham
,
used
velocity-
and
direction-dependent
mass
concepts
in
his
1905
electrodynamics
paper
and
in
another
paper
in
1906.
<
ref
>
{
{
Cite
journal|last=Einstein|first=A.|date=1905|title=Zur
Elektrodynamik
bewegter
Körper|journal=Annalen
der
Physik|language=de|volume=322|issue=10|pages=891–921|doi=10.1002/andp.19053221004|bibcode=1905AnP
...
322..891E|trans-title=
[
http
:
//www.fourmilab.ch/etexts/einstein/specrel/www/
On
the
Electrodynamics
of
Moving
Bodies
]
|doi-access=free
}
}
<
/ref
>
<
ref
>
{
{
Cite
journal|last=Einstein|first=A.|date=1906|title=Über
eine
Methode
zur
Bestimmung
des
Verhältnisses
der
transversalen
und
longitudinalen
Masse
des
Elektrons|url=http
:
//doi.wiley.com/10.1002/andp.19063261310|journal=Annalen
der
Physik|language=de|volume=326|issue=13|pages=583–586|doi=10.1002/andp.19063261310|bibcode=1906AnP
...
326..583E|trans-title=
[
https
:
//einsteinpapers.press.princeton.edu/vol2-trans/221
On
a
method
for
the
determination
of
the
ratio
of
the
transverse
and
the
longitudinal
mass
of
the
electron
]
|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080247/https
:
//onlinelibrary.wiley.com/doi/abs/10.1002/andp.19063261310|url-status=live
}
}
<
/ref
>
In
Einstein
's
first
1905
paper
on
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
,
he
treated
{
{
mvar|m
}
}
as
what
would
now
be
called
the
``
rest
mass
''
,
<
ref
name=
''
inertia
''
/
>
and
it
has
been
noted
that
in
his
later
years
he
did
not
like
the
idea
of
``
relativistic
mass
''
.
<
ref
name=Okun
>
{
{
Cite
journal|last=Okun|first=Lev
B.|date=June
1989|title=The
Concept
of
Mass|url=http
:
//dx.doi.org/10.1063/1.881171|journal=Physics
Today|volume=42|issue=6|pages=31–36|doi=10.1063/1.881171|bibcode=1989PhT
...
.42f..31O|issn=0031-9228
}
}
<
/ref
>
In
older
physics
terminology
,
relativistic
energy
is
used
in
lieu
of
relativistic
mass
and
the
term
``
mass
''
is
reserved
for
the
rest
mass.
<
ref
name=elementaryParticles
/
>
Historically
,
there
has
been
considerable
debate
over
the
use
of
the
concept
of
``
relativistic
mass
''
and
the
connection
of
``
mass
''
in
relativity
to
``
mass
''
in
Newtonian
dynamics
.
One
view
is
that
only
rest
mass
is
a
viable
concept
and
is
a
property
of
the
particle
;
while
relativistic
mass
is
a
conglomeration
of
particle
properties
and
properties
of
spacetime
.
Another
view
,
attributed
to
Norwegian
physicist
Kjell
Vøyenli
,
is
that
the
Newtonian
concept
of
mass
as
a
particle
property
and
the
relativistic
concept
of
mass
have
to
be
viewed
as
embedded
in
their
own
theories
and
as
having
no
precise
connection.
<
ref
name=
''
Jammer
''
>
{
{
Cite
book|last=Jammer|first=Max|url=
https
:
//www.worldcat.org/oclc/614715841|title=Concepts
of
mass
in
contemporary
physics
and
philosophy|date=2000|publisher=Princeton
University
Press|isbn=978-1-4008-1219-6
|page=51|location=Princeton
,
N.J.|oclc=614715841
}
}
<
/ref
>
<
ref
name=
''
Vøyenli
''
>
{
{
Cite
journal|last1=Eriksen|first1=Erik|last2=Vøyenli|first2=Kjell|date=February
1976|title=The
classical
and
relativistic
concepts
of
mass|url=http
:
//link.springer.com/10.1007/BF00708670|journal=Foundations
of
Physics|language=en|volume=6|issue=1|pages=115–124|doi=10.1007/BF00708670|bibcode=1976FoPh
...
.6..115E|s2cid=120139174|issn=0015-9018
}
}
<
/ref
>
====Einstein
's
1905
derivation====
Already
in
his
relativity
paper
``
On
the
electrodynamics
of
moving
bodies
''
,
Einstein
derived
the
correct
expression
for
the
kinetic
energy
of
particles
:
:
$
E
_
k
=
m
_0
c
^2( \
gamma
-1 ) =
m
_0
c
^2\left(\frac{1}{\sqrt{1-\frac{
v
^2}{
c
^2}}} - 1\right),
$
.
Now
the
question
remained
open
as
to
which
formulation
applies
to
bodies
at
rest
.
This
was
tackled
by
Einstein
in
his
paper
``
Does
the
inertia
of
a
body
depend
upon
its
energy
content
?
``
,
one
of
his
[
[
Annus
Mirabilis
papers
]
]
.
Here
,
Einstein
used
{
{
math|
''
V
''
}
}
to
represent
the
speed
of
light
in
vacuum
and
{
{
math|
''
L
''
}
}
to
represent
the
[
[
energy
]
]
lost
by
a
body
in
the
form
of
radiation.
<
ref
name=
''
inertia
''
/
>
Consequently
,
the
equation
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
was
not
originally
written
as
a
formula
but
as
a
sentence
in
German
saying
that
``
if
a
body
gives
off
the
energy
{
{
math|
''
L
''
}
}
in
the
form
of
radiation
,
its
mass
diminishes
by
{
{
math|
{
{
sfrac|
''
L
''
|
''
V
''
<
sup
>
2
<
/sup
>
}
}
}
}
.
''
A
remark
placed
above
it
informed
that
the
equation
was
approximated
by
neglecting
``
magnitudes
of
fourth
and
higher
orders
''
of
a
[
[
Series
(
mathematics
)
|series
expansion
]
]
.
<
ref
group=note
>
See
the
sentence
on
the
last
page
641
of
the
original
German
edition
,
above
the
equation
{
{
math|1=
''
K
''
<
sub
>
0
<
/sub
>
−
``
K
''
<
sub
>
1
<
/sub
>
=
{
{
sfrac|
''
L
''
|
''
V
''
<
sup
>
2
<
/sup
>
}
}
{
{
sfrac|
''
v
''
<
sup
>
2
<
/sup
>
|2
}
}
}
}
.
See
also
the
sentence
above
the
last
equation
in
the
English
translation
,
{
{
math|1=
''
K
''
<
sub
>
0
<
/sub
>
−
``
K
''
<
sub
>
1
<
/sub
>
=
{
{
sfrac|1|2
}
}
(
{
{
sfrac|
''
L
''
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
)
''
v
''
<
sup
>
2
<
/sup
>
}
}
,
and
the
comment
on
the
symbols
used
in
``
About
this
edition
''
that
follows
the
translation.
<
/ref
>
Einstein
used
a
body
emitting
two
light
pulses
in
opposite
directions
,
having
energies
of
{
{
math|
''
E
''
<
sub
>
0
<
/sub
>
}
}
before
and
{
{
math|
''
E
''
<
sub
>
1
<
/sub
>
}
}
after
the
emission
as
seen
in
its
rest
frame
.
As
seen
from
a
moving
frame
,
this
becomes
{
{
math|
''
H
''
<
sub
>
0
<
/sub
>
}
}
and
{
{
math|
''
H
''
<
sub
>
1
<
/sub
>
}
}
.
Einstein
obtained
,
in
modern
notation
:
:
$
E
_
k
= \frac{1}{2}
m
_0
v
^2 + \cdots
$
.
He
then
argued
that
{
{
math|
''
H
''
−
``
E
''
}
}
can
only
differ
from
the
kinetic
energy
{
{
math|
''
K
''
}
}
by
an
additive
constant
,
which
gives
:
$
E
_{
k
}=
m
c
^{2}\left(\frac{1}{\sqrt{1-\frac{
v
^{2}}{
c
^{2}}}}-1\right)
$
.
Neglecting
effects
higher
than
third
order
in
{
{
math|
{
{
sfrac|
''
v
''
|
''
c
''
}
}
}
}
after
a
[
[
Taylor
series
]
]
expansion
of
the
right
side
of
this
yields
:
:
$
\left(
H
_{0}-
E
_{0}\right)-\left(
H
_{1}-
E
_{1}\right)=
E
\left(\frac{1}{\sqrt{1-\frac{
v
^{2}}{
c
^{2}}}}-1\right)
$
Einstein
concluded
that
the
emission
reduces
the
body
's
mass
by
{
{
math|
{
{
sfrac|
''
E
''
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
,
and
that
the
mass
of
a
body
is
a
measure
of
its
energy
content
.
The
correctness
of
Einstein
's
1905
derivation
of
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
was
criticized
by
German
theoretical
physicist
[
[
Max
Planck
]
]
in
1907
,
who
argued
that
it
is
only
valid
to
first
approximation
.
Another
criticism
was
formulated
by
American
physicist
[
[
Herbert
Ives
]
]
in
1952
and
the
Israeli
physicist
[
[
Max
Jammer
]
]
in
1961
,
asserting
that
Einstein
's
derivation
is
based
on
[
[
begging
the
question
]
]
.
<
ref
name=jammer2
>
{
{
Cite
book|last=Jammer|first=Max|url=https
:
//www.worldcat.org/oclc/37546758|title=Concepts
of
mass
:
in
classical
and
modern
physics|orig-year=1961|date=1997|publisher=Dover
Publications|isbn=978-0-486-29998-3|location=Mineola
,
N.Y.|oclc=37546758|page=51
}
}
<
/ref
>
<
ref
>
{
{
Cite
journal|last=Ives|first=Herbert
E.|date=1952-08-01|title=
Derivation
of
the
Mass-Energy
Relation|url=https
:
//www.osapublishing.org/abstract.cfm
?
URI=josa-42-8-540|journal=Journal
of
the
Optical
Society
of
America|language=en|volume=42|issue=8
|page=540|doi=10.1364/JOSA.42.000540
|issn=0030-3941
}
}
<
/ref
>
Other
scholars
,
such
as
American
and
Chilean
[
[
philosopher
]
]
s
[
[
John
Stachel
]
]
and
[
[
Roberto
Torretti
]
]
,
have
argued
that
Ives
'
criticism
was
wrong
,
and
that
Einstein
's
derivation
was
correct.
<
ref
>
{
{
Cite
journal|last1=Stachel|first1=John|last2=Torretti|first2=Roberto|date=August
1982|title=Einstein
's
first
derivation
of
mass–energy
equivalence|url=http
:
//aapt.scitation.org/doi/10.1119/1.12764|journal=American
Journal
of
Physics|language=en|volume=50|issue=8|pages=760–763|doi=10.1119/1.12764|bibcode=1982AmJPh..50..760S|issn=0002-9505|access-date=2020-10-14|archive-date=2019-05-28|archive-url=https
:
//web.archive.org/web/20190528172007/https
:
//aapt.scitation.org/doi/10.1119/1.12764|url-status=live
}
}
<
/ref
>
American
physics
writer
[
[
Hans
Ohanian
]
]
,
in
2008
,
agreed
with
Stachel/Torretti
's
criticism
of
Ives
,
though
he
argued
that
Einstein
's
derivation
was
wrong
for
other
reasons.
<
ref
>
{
{
Cite
journal|last=Ohanian|first=Hans
C.|date=May
2009|title=Did
Einstein
prove
E=mc2
?
|url=https
:
//linkinghub.elsevier.com/retrieve/pii/S1355219809000112|journal=Studies
in
History
and
Philosophy
of
Science
Part
B
:
Studies
in
History
and
Philosophy
of
Modern
Physics|language=en|volume=40|issue=2|pages=167–173|doi=10.1016/j.shpsb.2009.03.002|bibcode=2009SHPMP..40..167O|access-date=2020-10-14
}
}
<
/ref
>
====Relativistic
center-of-mass
theorem
of
1906====
Like
Poincaré
,
Einstein
concluded
in
1906
that
the
inertia
of
electromagnetic
energy
is
a
necessary
condition
for
the
center-of-mass
theorem
to
hold
.
On
this
occasion
,
Einstein
referred
to
Poincaré
's
1900
paper
and
wrote
:
``
Although
the
merely
formal
considerations
,
which
we
will
need
for
the
proof
,
are
already
mostly
contained
in
a
work
by
H.
Poincaré
<
sup
>
2
<
/sup
>
,
for
the
sake
of
clarity
I
will
not
rely
on
that
work
.
``
<
ref
>
{
{
Cite
journal|last=Einstein|first=A.|date=1906|title=Das
Prinzip
von
der
Erhaltung
der
Schwerpunktsbewegung
und
die
Trägheit
der
Energie|trans-title=
[
https
:
//einsteinpapers.press.princeton.edu/vol2-trans/214
The
Principle
of
Conservation
of
Motion
of
the
Center
of
Gravity
and
the
Inertia
of
Energy
]
|url=http
:
//doi.wiley.com/10.1002/andp.19063250814|journal=Annalen
der
Physik|language=de|volume=325|issue=8|pages=627–633|doi=10.1002/andp.19063250814|bibcode=1906AnP
...
325..627E|s2cid=120361282
|quote=Trotzdem
die
einfachen
formalen
Betrachtungen
,
die
zum
Nachweis
dieser
Behauptung
durchgeführt
werden
müssen
,
in
der
Hauptsache
bereits
in
einer
Arbeit
von
H.
Poincaré
enthalten
sind
<
sup
>
2
<
/sup
>
,
werde
ich
mich
doch
der
Übersichtlichkeit
halber
nicht
auf
jene
Arbeit
stützen.|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080259/https
:
//onlinelibrary.wiley.com/doi/abs/10.1002/andp.19063250814|url-status=live
}
}
<
/ref
>
In
Einstein
's
more
physical
,
as
opposed
to
formal
or
mathematical
,
point
of
view
,
there
was
no
need
for
fictitious
masses
.
He
could
avoid
the
``
[
[
perpetual
motion
]
]
''
problem
because
,
on
the
basis
of
the
mass–energy
equivalence
,
he
could
show
that
the
transport
of
inertia
that
accompanies
the
emission
and
absorption
of
radiation
solves
the
problem
.
Poincaré
's
rejection
of
the
principle
of
action–reaction
can
be
avoided
through
Einstein
's
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
,
because
mass
conservation
appears
as
a
special
case
of
the
[
[
energy
conservation
law
]
]
.
====Further
developments====
There
were
several
further
developments
in
the
first
decade
of
the
twentieth
century
.
In
May
1907
,
Einstein
explained
that
the
expression
for
energy
{
{
math|
''
ε
''
}
}
of
a
moving
mass
point
assumes
the
simplest
form
when
its
expression
for
the
state
of
rest
is
chosen
to
be
{
{
math|1=
''
ε
''
<
sub
>
0
<
/sub
>
=
``
μV
''
<
sup
>
2
<
/sup
>
}
}
(
where
{
{
math|
''
μ
''
}
}
is
the
mass
)
,
which
is
in
agreement
with
the
``
principle
of
the
equivalence
of
mass
and
energy
''
.
In
addition
,
Einstein
used
the
formula
{
{
math|1=
''
μ
''
=
{
{
sfrac|
''
E
''
<
sub
>
0
<
/sub
>
|
''
V
''
<
sup
>
2
<
/sup
>
}
}
}
}
,
with
{
{
math|
''
E
''
<
sub
>
0
<
/sub
>
}
}
being
the
energy
of
a
system
of
mass
points
,
to
describe
the
energy
and
mass
increase
of
that
system
when
the
velocity
of
the
differently
moving
mass
points
is
increased.
<
ref
>
{
{
Cite
journal|last=Einstein|first=A.|date=1907|title=Über
die
vom
Relativitätsprinzip
geforderte
Trägheit
der
Energie|trans-title=
[
https
:
//einsteinpapers.press.princeton.edu/vol2-trans/252
On
the
Inertial
of
Energy
Required
by
the
Relativity
Principle
]
|url=http
:
//doi.wiley.com/10.1002/andp.19073280713|journal=Annalen
der
Physik|language=de|volume=328|issue=7|pages=371–384|doi=10.1002/andp.19073280713|bibcode=1907AnP
...
328..371E
}
}
<
/ref
>
Max
Planck
rewrote
Einstein
's
mass–energy
relationship
as
{
{
math|1=
''
M
''
=
{
{
sfrac|
''
E
''
<
sub
>
0
<
/sub
>
+
``
pV
''
<
sub
>
0
<
/sub
>
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
in
June
1907
,
where
{
{
math|
''
p
''
}
}
is
the
pressure
and
{
{
math|
''
V
''
<
sub
>
0
<
/sub
>
}
}
the
volume
to
express
the
relation
between
mass
,
its
latent
energy
,
and
thermodynamic
energy
within
the
body.
<
ref
>
{
{
Cite
journal|last=Planck|first=M.|date=1908|title=Zur
Dynamik
bewegter
Systeme|trans-title=
[
[
s
:
Translation
:
On
the
Dynamics
of
Moving
Systems|On
the
Dynamics
of
Moving
Systems
]
]
|url=http
:
//doi.wiley.com/10.1002/andp.19083310602|journal=Annalen
der
Physik|language=de|volume=331|issue=6|pages=1–34|doi=10.1002/andp.19083310602|bibcode=1908AnP
...
331
...
.1P
}
}
<
/ref
>
Subsequently
,
in
October
1907
,
this
was
rewritten
as
{
{
math|1=
''
M
''
<
sub
>
0
<
/sub
>
=
{
{
sfrac|
''
E
''
<
sub
>
0
<
/sub
>
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
and
given
a
quantum
interpretation
by
German
physicist
[
[
Johannes
Stark
]
]
,
who
assumed
its
validity
and
correctness.
<
ref
>
{
{
cite
journal|author=Stark
,
J.|url=https
:
//archive.org/details/physikalischeze00unkngoog|title=Elementarquantum
der
Energie
,
Modell
der
negativen
und
der
positiven
Elekrizitat|journal=Physikalische
Zeitschrift|page=881|volume=24|issue=
8
|date=1907|language=de
}
}
<
/ref
>
In
December
1907
,
Einstein
expressed
the
equivalence
in
the
form
{
{
math|1=
''
M
''
=
``
μ
''
+
{
{
sfrac|
''
E
''
<
sub
>
0
<
/sub
>
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
and
concluded
:
``
A
mass
{
{
math|
''
μ
''
}
}
is
equivalent
,
as
regards
inertia
,
to
a
quantity
of
energy
{
{
math|
''
μc
<
sup
>
2
<
/sup
>
''
}
}
.
[
…
]
It
appears
far
more
natural
to
consider
every
inertial
mass
as
a
store
of
energy
.
``
<
ref
>
{
{
Cite
journal|last=Einstein|first=Albert|date=1908|title=Über
das
Relativitätsprinzip
und
die
aus
demselben
gezogenen
Folgerungen|url=http
:
//adsabs.harvard.edu/abs/1908JRE
...
..4..411E|trans-title=
[
https
:
//einsteinpapers.press.princeton.edu/vol2-trans/266
?
ajax
On
the
Relativity
Principle
and
the
Conclusions
Drawn
From
it
]
|journal=Jahrbuch
der
Radioaktivität
und
Elektronik|language=de|volume=4|page=411|bibcode=1908JRE
...
..4..411E
}
}
<
/ref
>
<
ref
>
{
{
Cite
journal|last=Schwartz|first=H
.
M.|date=September
1977|title=Einstein
's
comprehensive
1907
essay
on
relativity
,
part
II|url=http
:
//aapt.scitation.org/doi/10.1119/1.11053|journal=American
Journal
of
Physics|language=en|volume=45|issue=9|pages=811–817|doi=10.1119/1.11053|bibcode=1977AmJPh..45..811S|issn=0002-9505|access-date=2020-10-14|archive-date=2019-05-28|archive-url=https
:
//web.archive.org/web/20190528171924/https
:
//aapt.scitation.org/doi/10.1119/1.11053|url-status=live
}
}
<
/ref
>
American
[
[
physical
chemist
]
]
s
[
[
Gilbert
N.
Lewis
]
]
and
[
[
Richard
C.
Tolman
]
]
used
two
variations
of
the
formula
in
1909
:
{
{
math|1=
''
m
''
=
{
{
sfrac|
''
E
''
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
and
{
{
math|1=
''
m
''
<
sub
>
0
<
/sub
>
=
{
{
sfrac|
''
E
''
<
sub
>
0
<
/sub
>
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
,
with
{
{
mvar|E
}
}
being
the
relativistic
energy
(
the
energy
of
an
object
when
the
object
is
moving
)
,
{
{
math|
''
E
''
<
sub
>
0
<
/sub
>
}
}
is
the
rest
energy
(
the
energy
when
not
moving
)
,
{
{
mvar|m
}
}
is
the
relativistic
mass
(
the
rest
mass
and
the
extra
mass
gained
when
moving
)
,
and
{
{
math|
''
m
''
<
sub
>
0
<
/sub
>
}
}
is
the
rest
mass.
<
ref
>
{
{
Cite
journal|last1=Lewis|first1=Gilbert
N.|last2=Tolman|first2=Richard
C.|date=1909|title=The
Principle
of
Relativity
,
and
Non-Newtonian
Mechanics|url=https
:
//www.jstor.org/stable/20022495|journal=Proceedings
of
the
American
Academy
of
Arts
and
Sciences|language=en|volume=44|issue=25|page=711|doi=10.2307/20022495|jstor=20022495
}
}
<
/ref
>
The
same
relations
in
different
notation
were
used
by
Lorentz
in
1913
and
1914
,
though
he
placed
the
energy
on
the
left-hand
side
:
{
{
math|1=
''
ε
''
=
``
Mc
''
<
sup
>
2
<
/sup
>
}
}
and
{
{
math|1=
''
ε
''
<
sub
>
0
<
/sub
>
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
,
with
{
{
mvar|ε
}
}
being
the
total
energy
(
rest
energy
plus
kinetic
energy
)
of
a
moving
material
point
,
{
{
math|
''
ε
''
<
sub
>
0
<
/sub
>
}
}
its
rest
energy
,
{
{
mvar|M
}
}
the
relativistic
mass
,
and
{
{
mvar|m
}
}
the
invariant
mass.
<
ref
>
{
{
Cite
book|last=Lorentz|first=Hendrik
Antoon|url=https
:
//books.google.com/books
?
id=89PPAAAAMAAJ|title=Das
Relativitätsprinzip
:
drei
Vorlesungen
gehalten
in
Teylers
Stiftung
zu
Haarlem|date=1914|publisher=B.G
.
Teubner|language=de|trans-title=The
principle
of
relativity
:
three
lectures
given
in
Teyler
's
foundation
in
Haarlem|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080259/https
:
//books.google.com/books
?
id=89PPAAAAMAAJ|url-status=live
}
}
<
/ref
>
In
1911
,
German
physicist
[
[
Max
von
Laue
]
]
gave
a
more
comprehensive
proof
of
{
{
math|1=
''
M
''
<
sub
>
0
<
/sub
>
=
{
{
sfrac|
''
E
''
<
sub
>
0
<
/sub
>
|
''
c
''
<
sup
>
2
<
/sup
>
}
}
}
}
from
the
[
[
stress–energy
tensor
]
]
,
<
ref
>
{
{
Cite
journal|last=Laue|first=M.|date=1911|title=Zur
Dynamik
der
Relativitätstheorie|trans-title=
[
[
s
:
Translation
:
On
the
Dynamics
of
the
Theory
of
Relativity|On
the
Dynamics
of
the
Theory
of
Relativity
]
]
|url=http
:
//doi.wiley.com/10.1002/andp.19113400808|journal=Annalen
der
Physik|language=de|volume=340|issue=8|pages=524–542|doi=10.1002/andp.19113400808|bibcode=1911AnP
...
340..524L
}
}
<
/ref
>
which
was
later
generalized
by
German
mathematician
[
[
Felix
Klein
]
]
in
1918.
<
ref
>
{
{
Citation|author=Klein
,
Felix|title=Über
die
Integralform
der
Erhaltungssätze
und
die
Theorie
der
räumlich-geschlossenen
Welt|trans-title=On
the
integral
form
of
the
conservation
laws
and
the
theory
of
the
spatially
closed
world|journal=Göttinger
Nachrichten|date=1918|pages=394–423|url=http
:
//gdz.sub.uni-goettingen.de/dms/load/img/
?
PPN=PPN243240503
&
DMDID=DMDLOG_0051
}
}
<
/ref
>
Einstein
returned
to
the
topic
once
again
after
[
[
World
War
II
]
]
and
this
time
he
wrote
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
in
the
title
of
his
article
<
ref
>
{
{
cite
magazine|last=Einstein|first=A.|date=April
1946|title=
{
{
math|1=E
=
mc
<
sup
>
2
<
/sup
>
}
}
:
the
most
urgent
problem
of
our
time|url=http
:
//alberteinstein.info/vufind1/Record/EAR000034164|magazine=
[
[
Science
Illustrated
]
]
|publisher=
[
[
Bonnier
Publications
International
]
]
|volume=1|issue=1|at=item
417
in
the
``
Bibliography
''
.
pp
.
16-17|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080301/http
:
//alberteinstein.info/vufind1/Record/EAR000034164|url-status=live
}
}
<
/ref
>
intended
as
an
explanation
for
a
general
reader
by
analogy.
<
ref
>
{
{
Cite
book|last=Schilpp
,
Paul
Arthur|url=https
:
//www.worldcat.org/oclc/134995|title=Albert
Einstein
:
philosopher-scientist.|date=1970|publisher=Open
Court|isbn=978-0-87548-286-6|edition=3d
|location=La
Salle
,
Ill.|at=M.C
.
Shields
``
Bibliography
of
the
Writings
of
Albert
Einstein
to
May
1951
''
|oclc=134995
}
}
<
/ref
>
====Alternative
version====
An
alternative
version
of
Einstein
's
[
[
thought
experiment
]
]
was
proposed
by
American
theoretical
physicist
[
[
Fritz
Rohrlich
]
]
in
1990
,
who
based
his
reasoning
on
the
[
[
Doppler
effect
]
]
.
<
ref
>
{
{
Cite
journal|last=Rohrlich|first=Fritz|date=April
1990|title=An
elementary
derivation
of
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
|url=http
:
//aapt.scitation.org/doi/10.1119/1.16168|journal=American
Journal
of
Physics|language=en|volume=58|issue=4|pages=348–349|doi=10.1119/1.16168|issn=0002-9505|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https
:
//web.archive.org/web/20210221080302/https
:
//aapt.scitation.org/doi/10.1119/1.16168|url-status=live
}
}
<
/ref
>
Like
Einstein
,
he
considered
a
body
at
rest
with
mass
{
{
mvar|M
}
}
.
If
the
body
is
examined
in
a
frame
moving
with
nonrelativistic
velocity
{
{
mvar|v
}
}
,
it
is
no
longer
at
rest
and
in
the
moving
frame
it
has
momentum
{
{
math|1=
''
P
''
=
``
Mv
''
}
}
.
Then
he
supposed
the
body
emits
two
pulses
of
light
to
the
left
and
to
the
right
,
each
carrying
an
equal
amount
of
energy
{
{
math|
{
{
sfrac|
''
E
''
|2
}
}
}
}
.
In
its
rest
frame
,
the
object
remains
at
rest
after
the
emission
since
the
two
beams
are
equal
in
strength
and
carry
opposite
momentum
.
However
,
if
the
same
process
is
considered
in
a
frame
that
moves
with
velocity
{
{
math|
''
v
''
}
}
to
the
left
,
the
pulse
moving
to
the
left
is
[
[
redshift
]
]
ed
,
while
the
pulse
moving
to
the
right
is
[
[
blue
shift
]
]
ed
.
The
blue
light
carries
more
momentum
than
the
red
light
,
so
that
the
momentum
of
the
light
in
the
moving
frame
is
not
balanced
:
the
light
is
carrying
some
net
momentum
to
the
right
.
The
object
has
not
changed
its
velocity
before
or
after
the
emission
.
Yet
in
this
frame
it
has
lost
some
right-momentum
to
the
light
.
The
only
way
it
could
have
lost
momentum
is
by
losing
mass
.
This
also
solves
Poincaré
's
radiation
paradox
.
The
velocity
is
small
,
so
the
right-moving
light
is
blueshifted
by
an
amount
equal
to
the
nonrelativistic
[
[
Doppler
shift
]
]
factor
{
{
math|1
−
{
{
sfrac|
''
v
''
|
''
c
''
}
}
}
}
.
The
momentum
of
the
light
is
its
energy
divided
by
{
{
mvar|c
}
}
,
and
it
is
increased
by
a
factor
of
{
{
math|
{
{
sfrac|
''
v
''
|
''
c
''
}
}
}
}
.
So
the
right-moving
light
is
carrying
an
extra
momentum
{
{
math|Δ
''
P
''
}
}
given
by
:
:
$
K
_{0}-
K
_{1}=
E
\left(\frac{1}{\sqrt{1-\frac{
v
^{2}}{
c
^{2}}}}-1\right)
$
The
left-moving
light
carries
a
little
less
momentum
,
by
the
same
amount
{
{
math|Δ
''
P
''
}
}
.
So
the
total
right-momentum
in
both
light
pulses
is
twice
{
{
math|Δ
''
P
''
}
}
.
This
is
the
right-momentum
that
the
object
lost
.
:
$
K
_{0}-
K
_{1}=\frac{
E
}{
c
^{2}}\frac{
v
^{2}}{2}.
$
The
momentum
of
the
object
in
the
moving
frame
after
the
emission
is
reduced
to
this
amount
:
:
$
\
Delta
P
= {
v
\over
c
}{
E
\over 2c} .
$
So
the
change
in
the
object
's
mass
is
equal
to
the
total
energy
lost
divided
by
{
{
math|
''
c
''
<
sup
>
2
<
/sup
>
}
}
.
Since
any
emission
of
energy
can
be
carried
out
by
a
two-step
process
,
where
first
the
energy
is
emitted
as
light
and
then
the
light
is
converted
to
some
other
form
of
energy
,
any
emission
of
energy
is
accompanied
by
a
loss
of
mass
.
Similarly
,
by
considering
absorption
,
a
gain
in
energy
is
accompanied
by
a
gain
in
mass
.
===Radioactivity
and
nuclear
energy===
[
[
File
:
Einstein
-
Time
Magazine
-
July
1
,
1946.jpg|right|thumb|The
popular
connection
between
Einstein
,
the
equation
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
,
and
the
[
[
nuclear
weapon|atomic
bomb
]
]
was
prominently
indicated
on
the
cover
of
``
[
[
Time
(
magazine
)
|Time
]
]
''
magazine
in
July
1946
.
]
]
It
was
quickly
noted
after
the
discovery
of
[
[
radioactivity
]
]
in
1897
that
the
total
energy
due
to
radioactive
processes
is
about
one
``
million
times
''
greater
than
that
involved
in
any
known
molecular
change
,
raising
the
question
of
where
the
energy
comes
from
.
After
eliminating
the
idea
of
absorption
and
emission
of
some
sort
of
Lesagian
ether
particles
,
the
existence
of
a
huge
amount
of
latent
energy
,
stored
within
matter
,
was
proposed
by
New
Zealand
physicist
[
[
Ernest
Rutherford
]
]
and
British
radiochemist
[
[
Frederick
Soddy
]
]
in
1903
.
Rutherford
also
suggested
that
this
internal
energy
is
stored
within
normal
matter
as
well
.
He
went
on
to
speculate
in
1904
:
``
If
it
were
ever
found
possible
to
control
at
will
the
rate
of
disintegration
of
the
radio-elements
,
an
enormous
amount
of
energy
could
be
obtained
from
a
small
quantity
of
matter
.
``
<
ref
>
{
{
Cite
book|last=Rutherford
,
Ernest|url=https
:
//www.worldcat.org/oclc/850842708|title=Radio-activity|date=2007|pages=336–338|publisher=Juniper
Grove|isbn=978-1-60355-058-1|edition=2nd|location=New
York|oclc=850842708
}
}
<
/ref
>
<
ref
>
{
{
Cite
book|last=Heisenberg|first=Werner|title=Physics
And
Philosophy
The
Revolution
In
Modern
Science|url=https
:
//archive.org/details/physicsandphilos010613mbp|date=1958|pages=118–119|publisher=HarperCollins|isbn=978-0-06-120919-2|language=en
}
}
<
/ref
>
Einstein
's
equation
does
not
explain
the
large
energies
released
in
radioactive
decay
,
but
can
be
used
to
quantify
them
.
The
theoretical
explanation
for
radioactive
decay
is
given
by
the
[
[
nuclear
force
]
]
s
responsible
for
holding
atoms
together
,
though
these
forces
were
still
unknown
in
1905
.
The
enormous
energy
released
from
radioactive
decay
had
previously
been
measured
by
Rutherford
and
was
much
more
easily
measured
than
the
small
change
in
the
gross
mass
of
materials
as
a
result
.
Einstein
's
equation
,
by
theory
,
can
give
these
energies
by
measuring
mass
differences
before
and
after
reactions
,
but
in
practice
,
these
mass
differences
in
1905
were
still
too
small
to
be
measured
in
bulk
.
Prior
to
this
,
the
ease
of
measuring
radioactive
decay
energies
with
a
[
[
calorimeter
]
]
was
thought
possibly
likely
to
allow
measurement
of
changes
in
mass
difference
,
as
a
check
on
Einstein
's
equation
itself
.
Einstein
mentions
in
his
1905
paper
that
mass–energy
equivalence
might
perhaps
be
tested
with
radioactive
decay
,
which
was
known
by
then
to
release
enough
energy
to
possibly
be
``
weighed
,
''
when
missing
from
the
system
.
However
,
radioactivity
seemed
to
proceed
at
its
own
unalterable
pace
,
and
even
when
simple
nuclear
reactions
became
possible
using
proton
bombardment
,
the
idea
that
these
great
amounts
of
usable
energy
could
be
liberated
at
will
with
any
practicality
,
proved
difficult
to
substantiate
.
Rutherford
was
reported
in
1933
to
have
declared
that
this
energy
could
not
be
exploited
efficiently
:
``
Anyone
who
expects
a
source
of
power
from
the
transformation
of
the
atom
is
talking
[
[
moonshine
]
]
.
``
<
ref
>
{
{
Cite
book|last=Reed|first=Bruce
Cameron|url=https
:
//books.google.com/books
?
id=RN5xCgAAQBAJ|title=Atomic
Bomb
:
The
Story
of
the
Manhattan
Project
:
How
nuclear
physics
became
a
global
geopolitical
game-changer|date=2015-06-01|publisher=Morgan
&
Claypool
Publishers|isbn=978-1-62705-992-3|language=en|section=The
neutrino
,
artificial
radioactivity
and
new
elements|quote=We
might
in
these
processes
obtain
very
much
more
energy
than
the
proton
supplied
,
but
on
the
average
we
could
not
expect
to
obtain
energy
in
this
way
.
It
was
a
very
poor
and
inefficient
way
of
producing
energy
,
and
anyone
who
looked
for
a
source
of
power
in
the
transformation
of
the
atoms
was
talking
moonshine
.
But
the
subject
was
scientifically
interesting
because
it
gave
insight
into
the
atoms.|at=Second
page
of
section
2.2
}
}
<
/ref
>
This
outlook
changed
dramatically
in
1932
with
the
discovery
of
the
neutron
and
its
mass
,
allowing
mass
differences
for
single
[
[
nuclide
]
]
s
and
their
reactions
to
be
calculated
directly
,
and
compared
with
the
sum
of
masses
for
the
particles
that
made
up
their
composition
.
In
1933
,
the
energy
released
from
the
reaction
of
[
[
lithium-7
]
]
plus
protons
giving
rise
to
two
[
[
alpha
particle
]
]
s
,
allowed
Einstein
's
equation
to
be
tested
to
an
error
of
±0.5
%
.
<
ref
>
{
{
cite
journal
|
last1
=
Oliphant
|first1=M
.
L.
E.
|
authorlink1
=
Mark
Oliphant
|
last2
=
Kinsey
|
first2
=
B
.
B
.
|
last3
=
Lord
Rutherford
|
authorlink3
=
Ernest
Rutherford
|
title
=
The
Transformation
of
Lithium
by
Protons
and
by
Ions
of
the
Heavy
Isotope
of
Hydrogen
|
year
=
1933
|
journal
=
Proceedings
of
the
Royal
Society
|
volume
=
141
|
issue
=
845
|
pages
=
722–733
|doi=10.1098/rspa.1933.0150
|s2cid=93342501
|
url
=
https
:
//royalsocietypublishing.org/doi/pdf/10.1098/rspa.1933.0150
}
}
<
/ref
>
However
,
scientists
still
did
not
see
such
reactions
as
a
practical
source
of
power
,
due
to
the
energy
cost
of
accelerating
reaction
particles
.
After
the
very
public
demonstration
of
huge
energies
released
from
nuclear
fission
after
the
[
[
atomic
bombings
of
Hiroshima
and
Nagasaki
]
]
in
1945
,
the
equation
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
became
directly
linked
in
the
public
eye
with
the
power
and
peril
of
nuclear
weapons
.
The
equation
was
featured
on
page
2
of
the
[
[
Smyth
Report
]
]
,
the
official
1945
release
by
the
US
government
on
the
development
of
the
atomic
bomb
,
and
by
1946
the
equation
was
linked
closely
enough
with
Einstein
's
work
that
the
cover
of
``
[
[
Time
(
magazine
)
|Time
]
]
''
magazine
prominently
featured
a
picture
of
Einstein
next
to
an
image
of
a
[
[
mushroom
cloud
]
]
emblazoned
with
the
equation.
<
ref
>
{
{
Cite
web|title=TIME
Magazine
--
U.S
.
Edition
--
July
1
,
1946
Vol
.
XLVIII
No
.
1|url=http
:
//content.time.com/time/magazine/0,9263,7601460701,00.html|access-date=2020-10-14|website=content.time.com|language=en-us|archive-date=2020-10-15|archive-url=https
:
//web.archive.org/web/20201015042338/http
:
//content.time.com/time/magazine/0,9263,7601460701,00.html|url-status=live
}
}
<
/ref
>
Einstein
himself
had
only
a
minor
role
in
the
[
[
Manhattan
Project
]
]
:
he
had
[
[
Einstein–Szilárd
letter|cosigned
a
letter
]
]
to
the
U.S.
president
in
1939
urging
funding
for
research
into
atomic
energy
,
warning
that
an
atomic
bomb
was
theoretically
possible
.
The
letter
persuaded
Roosevelt
to
devote
a
significant
portion
of
the
wartime
budget
to
atomic
research
.
Without
a
security
clearance
,
Einstein
's
only
scientific
contribution
was
an
analysis
of
an
[
[
isotope
separation
]
]
method
in
theoretical
terms
.
It
was
inconsequential
,
on
account
of
Einstein
not
being
given
sufficient
information
to
fully
work
on
the
problem.
<
ref
>
{
{
Cite
book|last=Isaacson
,
Walter|url=https
:
//www.worldcat.org/oclc/76961150|title=Einstein
:
his
life
and
universe|chapter=The
bomb|date=10
April
2007|isbn=978-0-7432-6473-0|location=New
York|oclc=76961150|access-date=14
October
2020|archive-date=22
August
2020|archive-url=https
:
//web.archive.org/web/20200822170147/http
:
//www.worldcat.org/oclc/76961150|url-status=live
}
}
<
/ref
>
While
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
is
useful
for
understanding
the
amount
of
energy
potentially
released
in
a
fission
reaction
,
it
was
not
strictly
necessary
to
develop
the
weapon
,
once
the
fission
process
was
known
,
and
its
energy
measured
at
200
&
nbsp
;
[
[
MeV
]
]
(
which
was
directly
possible
,
using
a
quantitative
[
[
Geiger
counter
]
]
,
at
that
time
)
.
The
physicist
and
Manhattan
Project
participant
[
[
Robert
Serber
]
]
noted
that
somehow
``
the
popular
notion
took
hold
long
ago
that
Einstein
's
theory
of
relativity
,
in
particular
his
famous
equation
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
,
plays
some
essential
role
in
the
theory
of
fission
.
Einstein
had
a
part
in
alerting
the
United
States
government
to
the
possibility
of
building
an
atomic
bomb
,
but
his
theory
of
relativity
is
not
required
in
discussing
fission
.
The
theory
of
fission
is
what
physicists
call
a
non-relativistic
theory
,
meaning
that
relativistic
effects
are
too
small
to
affect
the
dynamics
of
the
fission
process
significantly
.
``
{
{
refn|group=note|
{
{
Cite
book|last=Serber|first=Robert|url=http
:
//dx.doi.org/10.2307/j.ctvw1d5pf|title=The
Los
Alamos
Primer|date=2020-04-07|publisher=University
of
California
Press|doi=10.2307/j.ctvw1d5pf|isbn=978-0-520-37433-1|page=7|s2cid=91948043
}
}
.
The
quotation
is
taken
from
Serber
's
1992
version
,
and
is
not
in
the
original
1943
[
[
Los
Alamos
Primer
]
]
of
the
same
name
.
}
}
There
are
other
views
on
the
equation
's
importance
to
nuclear
reactions
.
In
late
1938
,
the
Austrian-Swedish
and
British
physicists
[
[
Lise
Meitner
]
]
and
[
[
Otto
Robert
Frisch
]
]
—while
on
a
winter
walk
during
which
they
solved
the
meaning
of
Hahn
's
experimental
results
and
introduced
the
idea
that
would
be
called
atomic
fission—directly
used
Einstein
's
equation
to
help
them
understand
the
quantitative
energetics
of
the
reaction
that
overcame
the
``
surface
tension-like
''
forces
that
hold
the
nucleus
together
,
and
allowed
the
fission
fragments
to
separate
to
a
configuration
from
which
their
charges
could
force
them
into
an
energetic
``
fission
''
.
To
do
this
,
they
used
``
packing
fraction
''
,
or
nuclear
[
[
binding
energy
]
]
values
for
elements
.
These
,
together
with
use
of
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
allowed
them
to
realize
on
the
spot
that
the
basic
fission
process
was
energetically
possible
.
{
{
refn|group=note|
{
{
block
quote|We
walked
up
and
down
in
the
snow
,
I
on
skis
and
she
on
foot…
and
gradually
the
idea
took
shape…
explained
by
Bohr
's
idea
that
the
nucleus
is
like
a
liquid
drop
;
such
a
drop
might
elongate
and
divide
itself…
We
knew
there
were
strong
forces
that
would
resist
,
..just
as
surface
tension
.
But
nuclei
differed
from
ordinary
drops
.
At
this
point
we
both
sat
down
on
a
tree
trunk
and
started
to
calculate
on
scraps
of
paper…
the
Uranium
nucleus
might
indeed
be
a
very
wobbly
,
unstable
drop
,
ready
to
divide
itself…
But…
when
the
two
drops
separated
they
would
be
driven
apart
by
electrical
repulsion
,
about
200
MeV
in
all
.
Fortunately
Lise
Meitner
remembered
how
to
compute
the
masses
of
nuclei…
and
worked
out
that
the
two
nuclei
formed…
would
be
lighter
by
about
one-fifth
the
mass
of
a
proton
.
Now
whenever
mass
disappears
energy
is
created
,
according
to
Einstein
's
formula
{
{
math|1=
''
E
''
=
``
mc
''
<
sup
>
2
<
/sup
>
}
}
,
and…
the
mass
was
just
equivalent
to
200
MeV
;
it
all
fitted
!
|author=Lise
Meitner
<
ref
>
{
{
Cite
book|last=Sime
,
Ruth
Lewin|url=https
:
//www.worldcat.org/oclc/42855101|title=Lise
Meitner
:
a
life
in
physics|date=1996|publisher=University
of
California
Press|isbn=978-0-520-91899-3|location=Berkeley|pages
=
236–237
|
oclc=42855101
}
}
<
/ref
>
}
}
}
}
===Einstein
's
equation
written===
According
to
the
Einstein
Papers
Project
at
the
[
[
California
Institute
of
Technology
]
]
and
[
[
Hebrew
University
of
Jerusalem
]
]
,
there
remain
only
four
known
copies
of
this
equation
as
written
by
Einstein
.
One
of
these
is
a
letter
written
in
[
[
German
language|German
]
]
to
[
[
Ludwik
Silberstein
]
]
,
which
was
in
Silberstein
's
archives
,
and
sold
at
[
[
auction
]
]
for
$
1.2
million
,
[
[
RR
Auction
]
]
of
[
[
Boston
,
Massachusetts
]
]
said
on
May
21
,
2021.
<
ref
>
{
{
cite
news
|
author
=
<
!
--
not
stated
--
>
|
title
=Handwritten
example
of
Einstein
equation
fetches
$
1.2M
|
url
=https
:
//apnews.com/article/lifestyle-science-4ed59f3bd9d9969354dd40ec363fe0e5
|
work
=Associated
Press
|
date
=May
21
,
2021
|
access-date
=April
11
,
2023
}
}
<
/ref
>
==See
also==
{
{
Portal|Physics
}
}
{
{
cmn|colwidth=30em|
*
[
[
Energy
density
]
]
*
[
[
Index
of
energy
articles
]
]
*
[
[
Index
of
wave
articles
]
]
*
[
[
Lorentz
transform
]
]
*
[
[
Length
contraction
]
]
*
[
[
Outline
of
energy
]
]
*
[
[
Relativity
of
simultaneity
]
]
}
}
==Notes==
{
{
Reflist|group=note
}
}
==References==
{
{
reflist
}
}
==External
links==
{
{
Wikisourcepar|Relativity
:
The
Special
and
General
Theory
}
}
{
{
Commons
category|Einstein
formula
}
}
*
[
http
:
//www.mathpages.com/home/kmath600/kmath600.htm
Einstein
on
the
Inertia
of
Energy
]
–
MathPages
*
[
https
:
//history.aip.org/exhibits/einstein/voice1_text.htm
Einstein-on
film
explaining
a
mass
energy
equivalence
]
*
[
http
:
//profmattstrassler.com/articles-and-posts/particle-physics-basics/mass-energy-matter-etc/mass-and-energy/
Mass
and
Energy
]
–
Conversations
About
Science
with
Theoretical
Physicist
Matt
Strassler
*
[
http
:
//plato.stanford.edu/entries/equivME
The
Equivalence
of
Mass
and
Energy
]
–
Entry
in
the
``
Stanford
Encyclopedia
of
Philosophy
''
*
{
{
cite
web|last=Merrifield|first=Michael|title=E=mc
<
sup
>
2
<
/sup
>
–
Mass–Energy
Equivalence|url=http
:
//www.sixtysymbols.com/videos/emc2.htm|work=Sixty
Symbols|publisher=
[
[
Brady
Haran
]
]
for
the
[
[
University
of
Nottingham
]
]
|author2=Copeland
,
Ed
|author3=Bowley
,
Roger|ref=none
}
}
{
{
Einstein
}
}
{
{
Relativity
}
}
{
{
DEFAULTSORT
:
Mass-Energy
Equivalence
}
}
[
[
Category:1905
introductions
]
]
[
[
Category:1905
in
science
]
]
[
[
Category:1905
in
Germany
]
]
[
[
Category
:
Albert
Einstein
]
]
[
[
Category
:
Energy
(
physics
)
]
]
[
[
Category
:
Equations
]
]
[
[
Category
:
Mass
]
]
[
[
Category
:
Special
relativity
]
]