Gravitational
Mass-Energy
Without
GR?
A
Food for Thought
Sir
Kevin Aylward B.Sc., Warden of the King’s Ale
Back
to the Contents section
Overview
The following is an attempt to extend on the idea of
the derivation of E=mc2 without using the Lorentz Transform, thus it is
recommended that that approach be first reviewed.
The approach used
here produces an equivalent result of mass variation for the instance of
gravitational potential energy rather than kinetic energy. The principle
illustrated is the often quoted result from general relativity, that winding up
a mechanical watch increases its mass by way of increasing its potential
energy.
Its intent is to give a clearer insight into the
approach of General Relativity where mass-energy is indeed rather flexible and
difficult to actually pin down. The
simplified argument used here, although interesting, does result in a contradiction
to General Relativity and experimental evidence. However, the road to truth
always takes many false paths on the way…
Derivation
The initial assumption is made that there is a small
test mass mr, that is
subject to a gravitational field from a much larger mass Mg such that any change in mass of the test mass mr as it is moved with
respect to the larger mass generating the gravitational field, is reflected in
the test mass and not the mass Mg
generating the potential. This assumption is very problematic, but allows an
initial simplified argument to be constructed. It is expected that
reformulating with both masses subject to variation would lead to an improved
result
This assumption alone illustrates the general problem
in General Relativity of identifying how to define mass. Mass is not a unique
property of an object, but a relationship between all other masses in the
universe. Mass can, in principle, be assigned in a somewhat arbitrary way
because energy and mass are equivalent.
The starting points are the Newtonian gravitational
equations:
F=
G
m
r
M
g
r
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2da9maalaaabaGaam4raiaad2gadaWgaaWcbaGaamOCaaqabaGccaWGnbWaaSbaaSqaaiaadEgaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaaaaa@3E75@
-
(1)
PE=
∫
F
dr
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpdaWdbaqaaiaadAeaaSqabeqaniabgUIiYdGccaWGKbGaamOCaaaa@3D2A@
-
(2)
If one assumes that the potential energy of a mass,
when it is moved in a gravitational potential, is contained by a change in its mass,
one can write, as for the kinetic energy case, in very general terms:
PE=
k
2
m(r)+α
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpcaWGRbWaaWbaaSqabeaacaaIYaaaaOGaamyBaiaacIcacaWGYbGaaiykaiabgUcaRiabeg7aHbaa@4021@
-
(3)
Where m(v), the mass, is an arbitrary function of position
r, k2 and α are arbitrary
constants.
One can write, simply to the KE example:
PE=
k
2
m(r)+α=
∫
G
m
r
M
g
r
2
dr
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpcaWGRbWaaWbaaSqabeaacaaIYaaaaOGaamyBaiaacIcacaWGYbGaaiykaiabgUcaRiabeg7aHjabg2da9maapeaabaWaaSaaaeaacaWGhbGaamyBamaaBaaaleaacaWGYbaabeaakiaad2eadaWgaaWcbaGaam4zaaqabaaakeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaaaeqabeqdcqGHRiI8aOGaamizaiaadkhaaaa@4BCF@
-
(4)
Differentiating both sides of the equation w.r.t:
k
2
m
'
(r)=
G
m
r
M
g
r
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCaaaleqabaGaaGOmaaaakiaad2gadaahaaWcbeqaaiaacEcaaaGccaGGOaGaamOCaiaacMcacqGH9aqpdaWcaaqaaiaadEeacaWGTbWaaSbaaSqaaiaadkhaaeqaaOGaamytamaaBaaaleaacaWGNbaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaaaa@43B1@
-
(5)
m
r
'
m
r
=
G
M
g
k
2
r
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGTbWaaSbaaSqaaiaadkhaaeqaaOWaaWbaaSqabeaacaGGNaaaaaGcbaGaamyBamaaBaaaleaacaWGYbaabeaaaaGccqGH9aqpdaWcaaqaaiaadEeacaWGnbWaaSbaaSqaaiaadEgaaeqaaaGcbaGaam4AamaaCaaaleqabaGaaGOmaaaakiaadkhadaahaaWcbeqaaiaaikdaaaaaaaaa@429E@
-
(6)
This equation can be immediately integrated:
ln(β
m
r
)=−
G
M
g
k
2
r
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGOaGaeqOSdiMaamyBamaaBaaaleaacaWGYbaabeaakiaacMcacqGH9aqpcqGHsisldaWcaaqaaiaadEeacaWGnbWaaSbaaSqaaiaadEgaaeqaaaGcbaGaam4AamaaCaaaleqabaGaaGOmaaaakiaadkhaaaaaaa@446F@
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35D5@
-
(7)
m
r
=
m
r0
e
−
G
M
g
k
2
r
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBaaaleaacaWGYbaabeaakiabg2da9iaad2gadaWgaaWcbaGaamOCaiaaicdaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0YaaSaaaeaacaWGhbGaamytamaaBaaameaacaWGNbaabeaaaSqaaiaadUgadaahaaadbeqaaiaaikdaaaWccaWGYbaaaaaaaaa@4385@
- (8)
Noting that at r = infinity, mr = mr0
Although it is not possible to show that k is c, the
speed of light in this derivation, for the purposes of this paper it will be
taken as such, so that:
m=
m
r0
e
−
G
M
g
c
2
r
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iaad2gadaWgaaWcbaGaamOCaiaaicdaaeqaaOGaamyzamaaCaaaleqabaGaeyOeI0YaaSaaaeaacaWGhbGaamytamaaBaaameaacaWGNbaabeaaaSqaaiaadogadaahaaadbeqaaiaaikdaaaWccaWGYbaaaaaaaaa@4250@
-
(9)
This equation holds that an object’s effective mass at
infinity, is increased as it gets further from a gravitational source. Substituting
into the force equation results in:
F=
G
M
g
m
r0
e
−
G
M
g
c
2
r
r
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2da9maalaaabaGaam4raiaad2eadaWgaaWcbaGaam4zaaqabaGccaWGTbWaaSbaaSqaaiaadkhacaaIWaaabeaakiaadwgadaahaaWcbeqaaiabgkHiTmaalaaabaGaam4raiaad2eadaWgaaadbaGaam4zaaqabaaaleaacaWGJbWaaWbaaWqabeaacaaIYaaaaSGaamOCaaaaaaaakeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaaaaa@46E3@
-
(10)
Which has the somewhat interesting property that in
the limit as r->0, the force is zero, due to the rate that the exponential
term goes to zero faster than the 1/r2 term does. Somewhat at odds
to the results of General Relativity’s black hole dynamics though
�
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FF@
….
The PE is given by integrating the force:
PE=−
c
2
m
r0
(1−
e
−
G
M
g
c
2
r
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpcqGHsislcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaamyBamaaBaaaleaacaWGYbGaaGimaaqabaGccaGGOaGaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTmaalaaabaGaam4raiaad2eadaWgaaadbaGaam4zaaqabaaaleaacaWGJbWaaWbaaWqabeaacaaIYaaaaSGaamOCaaaaaaGccaGGPaaaaa@48D0@
-
(11)
Substituting back into to - (3), with suitable initial
conditions will give:
PE=
c
2
(m−
m
o
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiaad2gacqGHsislcaWGTbWaaSbaaSqaaiaad+gaaeqaaOGaaiykaaaa@3FA9@
-
(12)
or
PE=
c
2
Δm
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaeuiLdqKaamyBaaaa@3CAD@
-
(13)
Which is in accord with the
results from General Relativity
Expanding the mass equation exponential
to first order results in:
m=
m
r0
(1−
G
M
g
c
2
r
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iaad2gadaWgaaWcbaGaamOCaiaaicdaaeqaaOGaaiikaiaaigdacqGHsisldaWcaaqaaiaadEeacaWGnbWaaSbaaSqaaiaadEgaaeqaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaakiaadkhaaaGaaiykaaaa@4349@
-
(14)
Expanding the potential
energy exponential to second order results in:
PE=−
c
2
m
r0
(1−(1−
G
M
g
c
2
r
+
1
2
(
G
M
g
c
2
r
)
2
))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpcqGHsislcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaamyBamaaBaaaleaacaWGYbGaaGimaaqabaGccaGGOaGaaGymaiabgkHiTiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaWGhbGaamytamaaBaaaleaacaWGNbaabeaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaGccaWGYbaaaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaam4raiaad2eadaWgaaWcbaGaam4zaaqabaaakeaacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaamOCaaaacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaaiykaiaacMcaaaa@5416@
-
(15)
PE=−
G
M
g
m
r0
r
(1−
G
M
g
2
c
2
r
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaadweacqGH9aqpcqGHsisldaWcaaqaaiaadEeacaWGnbWaaSbaaSqaaiaadEgaaeqaaOGaamyBamaaBaaaleaacaWGYbGaaGimaaqabaaakeaacaWGYbaaaiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaWGhbGaamytamaaBaaaleaacaWGNbaabeaaaOqaaiaaikdacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaamOCaaaacaGGPaaaaa@4966@
-
(16)
This is the somewhat problematic issue alluded to at
the introduction. The correction to the Newtonian potential energy is only
½
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=1laaaa@384B@
of
that required to account for gravitational redshift.
Looks like the original Einstein calculation that
resulted in
½
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=1laaaa@384B@
of
the correct value has popped up once again….
I will leave it as an exercise for the reader to
determine if allowing both masses to vary, fixes the problem….
© Kevin Aylward 2000 - 2023
All rights reserved
The information on the page may be
reproduced
providing that this source is acknowledged.
Website last modified 3rd December
2023
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