Special and General Relativity Lecture Notes:
Day 8 (09/16/08)
Shawn Mitryk
Contents
1 Stress Tensors
1.1 Previously Covered . . . . . . .
1.2 Single Particle Stress Tensor . .
1.3 Multiple Particle Stress Tensor
1.3.1 Projector . . . . . . . .
1.3.2 Imperfect Fluid Traits .
1.4 Stress Tensor Conservation . .
1.5 Definition of a Stress Tensor . .
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2
2
2
2
3
3
3
4
2 Parallel Transport
4
3 Next Class
3.1 Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
5
1
1
Stress Tensors
1.1
Previously Covered
• Scalar Field (φ)
∇µ T µν
∂V
∇µ ∇µ φ −
∂φ
=
0
(1)
=
0
(2)
• Electromagnetic (F µν )
1.2
Single Particle Stress Tensor
Z
µν
Ts.p.
=
uµ
=
muµ uν
δ 4 (xa − z a (τ ))
√
dτ
−g
dz µ (τ )
dτ
(3)
(4)
where z µ (τ ) is the orbit of the particle through space.
This can also be written as:
µν
Ts.p.
=
muµ uν
δ 3 (xa − z a (τ ))
|ż 0 (τ )|
(5)
This result depends only on the orbit of the particle and not on the spacetime curvature through which it moves
1.3
Multiple Particle Stress Tensor
P
Taking into account a group of particles given by i mi uµi uνi δ we can take the
continuous limit:
X
mi uµi uνi δ = ρuµ uν
(6)
i
Notice that the Energy Density is explicit in the stress tensor:
EnergyDensity
=
ρu0 u0
(7)
Now we can include particle-particle and particle-boundary interactions.
These give rise to a locally isotropic pressure in the spatial directions.
Thus in a co-moving frame:


ρu0 u0 0 0 0

0
p 0 0 


Tfµν
(8)
luid = 
0
0 p 0 
0
0 0 p
2
1.3.1
Projector
Basic Projector:
P µν
= uµ uν + g µν
(9)
Note: This tensor has rank 3 (degenerate; not maximal)
P µν uν
=
(uµ uν + g µν )uν = uµ − uµ = 0
(10)
Now, constructing the entire stress tensor for a Perfect Fluid:
T µν
T
1.3.2
µν
=
=
ρuµ uν + pP µν
µ ν
(11)
µ ν
ρu u + p(u u + g
µν
)
(12)
Imperfect Fluid Traits
• Finite Size
• Compressible
• Viscosity
• Dissipation
• Non-turbulent
• Finite heat capacity/conductivity
1.4
Stress Tensor Conservation
µν
∇µ T µν = T;µ
=
0
(13)
µ
=
0
(14)
a
Thus ”u” motion is geodesic.
The particle must move on a geodesic of the curvature that it is responsible
for.
Breaking the conservation equation into the scalar and spatial equations:
uν ∇µ T µν
=
0
(15)
Pνσ ∇µ T µν
=
0
(16)
The first of these reduces to the scalar Equation of continuity. The second
of these reduces to the vector Euler’s Equation for perfect fluids.
3
1.5
Definition of a Stress Tensor
A stress tensor T µν defines the pµ momentum transfer across a xν interface.
Note: T µν must be symmetric since it is constructed from the variational
principle δgδSµν which is itself symmetric.
2
Parallel Transport
Parallel Transport Equation given a vector V a
dV a
+ Γabc V b uc
dλ
=
0
(17)
uc
=
dxc
dλ
(18)
Given a flat space 2-D plane in polar coordinates we can write the invariant
length:
ds2
= dr2 + r2 dθ2
(19)
Noting the Christoff Coefficients:
Γrθθ
Γθrθ
= −r
1
=
r
(20)
(21)
Now assuming the velocity has one component in the θ direction [u =
dθ
(0, dλ
) = uθ ] we can write out:
dV r
dθ
− RV θ
dλ
dλ
dV θ
V r dθ
+
dλ
R dλ
=
0
(22)
=
0
(23)
Differentiating:
d2 V r
− Ruθ (dV θ dλ) = 0
dλ2
d2 V r
+ uθ2 V r = 0
dλ2
(24)
(25)
Applying I.C.s and Solving:
V r = V0r cos(uθ λ)
Vr
V θ = − 0 sin(uθ λ)
R
Noting that θ = uθ λ:
4
(26)
(27)
Solution becomes:
V r = V0r cos(θ)
Vr
V θ = − 0 sin(θ)
R
(28)
(29)
Thus there is no deficit angle.
One should notice that on the 2-sphere the Area is proportionate to the
deficit angle // (A = R2 δ) =⇒ Riemann Curvature ∝ R12 .
3
Next Class
3.1
Homework
• Problems 4, 7, 19, 23, 28, 30 from:
http://www.phys.ufl.edu/courses/phz6607/fall08/PC quest.pdf
• Find a definition of ”curvature”
3.2
Reading
• Finish Chapter 2 & 3 (on Expanding Universe)
5