General Relativity For
Tellytubbys
The Stress of Life That
causes One To Get Tense
Sir Kevin Aylward B.Sc.,
Warden of the Kings Ale
Back
to the Contents section
Overview
This section gives an
outline of the Stress-Energy or Energy-Momentum Tensor. This little
beastie is the thingymigigary that contains all the mass-energy and momentum of
the 3 universes, and more to boot.
Stress-Energy/Energy-Momentum
Tensor
The Stress Energy or Energy
Momentum Tensor is an object containing information about all the mass, energy,
and momentum of a system. Its covariant derivative results in the mass-energy
and momentum conservation equations, for example the mass flow continuity
equation and the Navier-Stokes equation of fluid mechanics all pop out in the
wash.
From the SR section, we have
The 4-position
X=[ct,x,y,z]=
x
α
e
α
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiwaiabg2da9iaacUfacaWGJbGaamiDaiaacYcacaWG4bGaaiilaiaadMhacaGGSaGaamOEaiaac2facqGH9aqpcaWG4bWaaWbaaSqabeaacqaHXoqyaaGccaWHLbWaaSbaaSqaaiabeg7aHbqabaaaaa@4718@
The 4-velocity
u=[cγ,γ
x
˙
,γ
y
˙
,γ
z
˙
]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2da9iaacUfacaWGJbGaeq4SdCMaaiilaiabeo7aNjqadIhagaGaaiaacYcacqaHZoWzceWG5bGbaiaacaGGSaGaeq4SdCMabmOEayaacaGaaiyxaaaa@4661@
u
0
≡c
d
x
0
dτ
=c
dt
dτ
=γc
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCaaaleqabaGaaGimaaaakiabggMi6kaadogadaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacaaIWaaaaaGcbaGaamizaiabes8a0baacqGH9aqpcaWGJbWaaSaaaeaacaWGKbGaamiDaaqaaiaadsgacqaHepaDaaGaeyypa0Jaeq4SdCMaam4yaaaa@4A48@
u
α
=
d
x
α
dτ
=γ
dx
dt
=γ
v
α
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCaaaleqabaGaeqySdegaaOGaeyypa0ZaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqySdegaaaGcbaGaamizaiabes8a0baacqGH9aqpcqaHZoWzdaWcaaqaaiaadsgacaWG4baabaGaamizaiaadshaaaGaeyypa0Jaeq4SdCMaamODamaaCaaaleqabaGaeqySdegaaaaa@4C3D@
,
alpha =1,2,3
The 4-momentum
p=mu
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiCaiabg2da9iaad2gacaWH1baaaa@39E3@
p=[
E
c
,γm
v
x
,γm
v
y
,γm
v
z
]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiCaiabg2da9iaacUfadaWcaaqaaiaadweaaeaacaWGJbaaaiaacYcacqaHZoWzcaWGTbGaamODamaaCaaaleqabaGaamiEaaaakiaacYcacqaHZoWzcaWGTbGaamODamaaCaaaleqabaGaamyEaaaakiaacYcacqaHZoWzcaWGTbGaamODamaaCaaaleqabaGaamOEaaaakiaac2faaaa@4BE0@
First, a refresh. This
assumes some prior fluid mechanics.
Given some dust collection
or fluid substance, with zero pressure, crossing some surface da it should be
seen that:
an element of mass flow in
unit time is dm=ρv.da
The total mass flow rate out
of a volume contained by that surface is therefore
m
˙
=
∬
ρv
.da
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaacaGaeyypa0Zaa8GaaeaacqaHbpGCcaWH2baaleqabeqdcqGHRiI8cqGHRiI8aOGaaiOlaiaahsgacaWHHbaaaa@410E@
The total mass in a volume
is given by
m=
∭
ρdv
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9maapmaabaGaeqyWdiNaamizaiaadAhaaSqabeqaniabgUIiYlabgUIiYlabgUIiYdaaaa@4123@
Therefore, what flows out
from the volume must equal what crosses the volume's enclosing surface.
∭
ρdv
=−
∯
a
ρv
.da
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8WaaeaacqaHbpGCcaWGKbGaamODaaWcbeqab0Gaey4kIiVaey4kIiVaey4kIipakiabg2da9iabgkHiTmaapyfabaGaeqyWdiNaaCODaaWcbaGaamyyaaqab0GaeSOeUlTaey4kIiVaey4kIipakiaac6cacaWHKbGaaCyyaaaa@4DDD@
And using Mr. Gauss or Mr. Green…
∂
∂t
∭
ρdv
=−
∭
∇.(ρv)
dv
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaamaapmaabaGaeqyWdiNaamizaiaadAhaaSqabeqaniabgUIiYlabgUIiYlabgUIiYdGccqGH9aqpcqGHsisldaWddaqaaiabgEGirlaah6cacaGGOaGaeqyWdiNaaCODaiaacMcaaSqabeqaniabgUIiYlabgUIiYlabgUIiYdGccaWGKbGaamODaaaa@52D3@
Hence:
∂ρ
∂t
+∇(ρv)=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcqaHbpGCaeaacqGHciITcaWG0baaaiabgUcaRiabgEGirlaacIcacqaHbpGCcaWH2bGaaiykaiabg2da9iaaicdaaaa@43C9@
This leads to a definition
of the energy-momentum tensor as the flux of 4-momemtum across a surface, but
this is far to complicated for us Teletubbys so lets wave a bit to Po.
Lets imagine a
parallelepiped (slant sided box) and its faces, with stuff flowing through the
faces. The force (vector) acting at any face will be a function of the area,
direction to that area and on object, that characterizes how all stuff is flowing
about. This object is the stress tensor i.e.
ΔF=σ.nΔa
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaaCOraiabg2da9iaaho8acaGGUaGaaCOBaiabfs5aejaadggaaaa@3E73@
So, one has an object that
is a product with the normal (vector) of the surface which must give a vector
as a result, therefor that object must be a tensor of rank 2
This can also be expressed
as
F=
∬
σ.nda
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOraiabg2da9maapiaabaaaleqabeqdcqGHRiI8cqGHRiI8aOGaaC4Wdiaac6cacaWHUbGaamizaiaadggaaaa@4061@
In component form we can
write
Δ
F
i
=
σ
ij
Δ
a
j
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaamOramaaCaaaleqabaGaamyAaaaakiabg2da9iabeo8aZnaaCaaaleqabaGaamyAaiaadQgaaaGccqqHuoarcaWGHbWaaSbaaSqaaiaadQgaaeqaaaaa@418E@
or as a definition of the
stress tensor
σ
ij
=
Δ
F
i
Δ
a
j
|
Δ
a
j
→0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaWGPbGaamOAaaaakiabg2da9maalaaabaGaeuiLdqKaamOramaaCaaaleqabaGaamyAaaaaaOqaaiabfs5aejaadggadaWgaaWcbaGaamOAaaqabaaaaOWaaqqaaeaadaWgaaWcbaGaeuiLdqKaamyyamaaBaaameaacaWGQbaabeaaliabgkziUkaaicdaaeqaaaGccaGLhWoaaaa@498C@
For our generic piece of
stuff floating around, let calculate in terms of momentum, cos we know what the
momentum density flow, per unit time, is from above, i.e. volume times density
is mass.
σ
ij
=
Δv
Δ
a
j
.
(ρ
v
i
)
Δt
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaWGPbGaamOAaaaakiabg2da9maalaaabaGaeuiLdqKaamODaaqaaiabfs5aejaadggadaahaaWcbeqaaiaadQgaaaaaaOGaaiOlamaalaaabaGaaiikaiabeg8aYjaadAhadaahaaWcbeqaaiaadMgaaaGccaGGPaaabaGaeuiLdqKaamiDaaaaaaa@490E@
σ
ij
=
Δ
x
i
Δ
x
j
Δ
x
k
Δ
a
j
.
(ρ
v
i
)
Δt
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaWGPbGaamOAaaaakiabg2da9maalaaabaGaeuiLdqKaamiEamaaCaaaleqabaGaamyAaaaakiabfs5aejaadIhadaahaaWcbeqaaiaadQgaaaGccqqHuoarcaWG4bWaaWbaaSqabeaacaWGRbaaaaGcbaGaeuiLdqKaamyyamaaCaaaleqabaGaamOAaaaaaaGccaGGUaWaaSaaaeaacaGGOaGaeqyWdiNaamODamaaCaaaleqabaGaamyAaaaakiaacMcaaeaacqqHuoarcaWG0baaaaaa@5148@
σ
ij
=
Δ
x
i
Δ
x
j
Δ
x
k
Δ
x
i
Δ
x
k
.
(ρ
v
i
)
Δt
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaWGPbGaamOAaaaakiabg2da9maalaaabaGaeuiLdqKaamiEamaaCaaaleqabaGaamyAaaaakiabfs5aejaadIhadaahaaWcbeqaaiaadQgaaaGccqqHuoarcaWG4bWaaWbaaSqabeaacaWGRbaaaaGcbaGaeuiLdqKaamiEamaaCaaaleqabaGaamyAaaaakiabfs5aejaadIhadaahaaWcbeqaaiaadUgaaaaaaOGaaiOlamaalaaabaGaaiikaiabeg8aYjaadAhadaahaaWcbeqaaiaadMgaaaGccaGGPaaabaGaeuiLdqKaamiDaaaaaaa@54E8@
σ
ij
=
Δ
x
j
Δt
.(ρ
v
i
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaWGPbGaamOAaaaakiabg2da9maalaaabaGaeuiLdqKaamiEamaaCaaaleqabaGaamOAaaaaaOqaaiabfs5aejaadshaaaGaaiOlaiaacIcacqaHbpGCcaWG2bWaaWbaaSqabeaacaWGPbaaaOGaaiykaaaa@46B4@
σ
ij
=
v
j
.(ρ
v
i
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaWGPbGaamOAaaaakiabg2da9iaadAhadaahaaWcbeqaaiaadQgaaaGccaGGUaGaaiikaiabeg8aYjaadAhadaahaaWcbeqaaiaadMgaaaGccaGGPaaaaa@42DD@
or
σ
ij
=ρ
v
i
v
j
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaWGPbGaamOAaaaakiabg2da9iabeg8aYjaadAhadaahaaWcbeqaaiaadMgaaaGccaWG2bWaaWbaaSqabeaacaWGQbaaaaaa@40C8@
which is just the tensor
product of velocities. i.e.
σ=ρv⊗v
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Wdiabg2da9iabeg8aYjaahAhacqGHxkcXcaWH2baaaa@3E10@
And now generalizing this
tensor to a 4-D space-time tensor we get
T=ρu⊗u
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCivaiabg2da9iabeg8aYjaahwhacqGHxkcXcaWH1baaaa@3D9C@
As the stress-energy or
energy-momentum, depending on which book you read, tensor
So, lets work out
T
0j
,j
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaCaaaleqabaGaaGimaiaadQgaaaGcdaWgaaWcbaGaaiilaiaadQgaaeqaaOGaeyypa0JaaGimaaaa@3C42@
(ρ
u
0
u
j
)
,j
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeg8aYjaadwhadaahaaWcbeqaaiaaicdaaaGccaWG1bWaaWbaaSqabeaacaWGQbaaaOGaaiykamaaBaaaleaacaGGSaGaamOAaaqabaGccqGH9aqpcaaIWaaaaa@40AD@
for low velocities, i.e.
gamma =1,
(ρ
u
0
u
j
)
,j
=(ρc
u
j
)
,
j
=0=(ρ
u
j
)
,
j
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeg8aYjaadwhadaahaaWcbeqaaiaaicdaaaGccaWG1bWaaWbaaSqabeaacaWGQbaaaOGaaiykamaaBaaaleaacaGGSaGaamOAaaqabaGccqGH9aqpcaGGOaGaeqyWdiNaam4yaiaadwhadaahaaWcbeqaaiaadQgaaaGccaGGPaGaaiilamaaBaaaleaacaWGQbaabeaakiabg2da9iaaicdacqGH9aqpcaGGOaGaeqyWdiNaamyDamaaCaaaleqabaGaamOAaaaakiaacMcacaGGSaWaaSbaaSqaaiaadQgaaeqaaaaa@51B3@
(ρc)
,0
+
(ρ
v
1
)
,1
+
(ρ
v
2
)
,2
+
(ρ
v
2
)
,3
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeg8aYjaadogacaGGPaWaaSbaaSqaaiaacYcacaaIWaaabeaakiabgUcaRiaacIcacqaHbpGCcaWG2bWaaWbaaSqabeaacaaIXaaaaOGaaiykamaaBaaaleaacaGGSaGaaGymaaqabaGccqGHRaWkcaGGOaGaeqyWdiNaamODamaaCaaaleqabaGaaGOmaaaakiaacMcadaWgaaWcbaGaaiilaiaaikdaaeqaaOGaey4kaSIaaiikaiabeg8aYjaadAhadaahaaWcbeqaaiaaikdaaaGccaGGPaWaaSbaaSqaaiaacYcacaaIZaaabeaakiabg2da9iaaicdaaaa@53F5@
∂(cρ)
∂
x
0
+
∂(ρ
v
1
)
∂
x
1
+
∂(ρ
v
2
)
∂
x
2
+
∂(ρ
v
3
)
∂
x
3
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@609E@
∂(cρ)
∂ct
+
∂(ρ
v
x
)
∂x
+
∂(ρ
v
y
)
∂y
+
∂(ρ
v
z
)
∂z
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaGGOaGaam4yaiabeg8aYjaacMcaaeaacqGHciITcaWGJbGaamiDaaaacqGHRaWkdaWcaaqaaiabgkGi2kaacIcacqaHbpGCcaWG2bWaaWbaaSqabeaacaWG4baaaOGaaiykaaqaaiabgkGi2kaadIhaaaGaey4kaSYaaSaaaeaacqGHciITcaGGOaGaeqyWdiNaamODamaaCaaaleqabaGaamyEaaaakiaacMcaaeaacqGHciITcaWG5baaaiabgUcaRmaalaaabaGaeyOaIyRaaiikaiabeg8aYjaadAhadaahaaWcbeqaaiaadQhaaaGccaGGPaaabaGaeyOaIyRaamOEaaaacqGH9aqpcaaIWaaaaa@5E81@
∂(ρ)
∂t
+∇.(ρv)=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaGGOaGaeqyWdiNaaiykaaqaaiabgkGi2kaadshaaaGaey4kaSIaey4bIeTaaiOlaiaacIcacqaHbpGCcaWH2bGaaiykaiabg2da9iaaicdaaaa@45D4@
So, one recovers the
conservation of mass equation.
It can also be seen from
inspection that the 0th row of the tensor contains the total energy and
momentum density of the stuff, i.e.
T
00
=ρ(γc)(γc)=
c
2
(ργ)γ=
E
dent
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaCaaaleqabaGaaGimaiaaicdaaaGccqGH9aqpcqaHbpGCcaGGOaGaeq4SdCMaam4yaiaacMcacaGGOaGaeq4SdCMaam4yaiaacMcacqGH9aqpcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiabeg8aYjabeo7aNjaacMcacqaHZoWzcqGH9aqpcaWGfbWaaSbaaSqaaiaadsgacaWGLbGaamOBaiaadshaaeqaaOGaaeiiaaaa@52BE@
Noting the contraction of
the volume element as well as the relativistic momentum density
T
0i
=c(ργ
v
i
)γ=c
p
i
where p
i
is the momentum density
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaCaaaleqabaGaaGimaiaadMgaaaGccqGH9aqpcaWGJbGaaiikaiabeg8aYjabeo7aNjaadAhadaahaaWcbeqaaiaadMgaaaGccaGGPaGaeq4SdCMaeyypa0Jaam4yaiaadchadaahaaWcbeqaaiaadMgaaaGccaqGGaGaae4DaiaabIgacaqGLbGaaeOCaiaabwgacaqGGaGaaeiCamaaCaaaleqabaGaaeyAaaaakiaabccacaqGPbGaae4CaiaabccacaqG0bGaaeiAaiaabwgacaqGGaGaaeyBaiaab+gacaqGTbGaaeyzaiaab6gacaqG0bGaaeyDaiaab2gacaqGGaGaaeizaiaabwgacaqGUbGaae4CaiaabMgacaqG0bGaaeyEaaaa@6486@
T
ij
=
σ
ij
= relativistic
f
i
a
j
which also is the flux of momentum in unit time accross a surface
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8950@
In general, with this
definition it can be shown that
∇.T=
T
ij
;j
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIeTaaiOlaiaahsfacqGH9aqpcaWGubWaaWbaaSqabeaacaWGPbGaamOAaaaakmaaBaaaleaacaGG7aGaamOAaaqabaGccqGH9aqpcaaIWaaaaa@40A0@
Perfect
Fluid
With a bit of piddling about
the energy momentum tensor for a perfect fluid can be derived as:
T=pg+(ρ+
p
c
2
)u⊗u
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCivaiabg2da9iaadchacaWHNbGaey4kaSIaaiikaiabeg8aYjabgUcaRmaalaaabaGaamiCaaqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaaiykaiaahwhacqGHxkcXcaWH1baaaa@457E@
Which, I will leave till
later.
© Kevin Aylward 2000 - 2022
All rights reserved
The information on the page may be
reproduced
providing that this source is acknowledged.
Website last modified 1st January
2022
http://www.kevinaylward.co.uk/gr/index.html
www.kevinaylward.co.uk