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Units, Constants and Useful Formulas
The Cauchy Stress Tensor
------------------------
Consider the stress, σ, inside a material.
The corresponding stress tensor is:
- -
| σ_{xx} σ_{xy} σ_{xz} |
σ = | σ_{yx} σ_{yy} σ_{yz} |
| σ_{zx} σ_{zy} σ_{zz} |
- -
Stress is a vector defined as σ = F/A which is the same formula
used to determine pressure. However, stress and pressure, though
related, are different. Pressure is a scalar since it is not
dependent on direction and is a force applied externally to an
object. Pressure causes stress inside of the object, so stress
is an internal force.
The pressure is the average of the 3 normal stresses at the
point, p = (1/3)Tr(σ). The off-diagonal components represent
shear stresses.
Stress-Energy-Momentum Tensor, T_{μν}
---------------------------------
The stress–energy-momentum tensor is a tensor that describes
the density and flux of energy and momentum in spacetime. It
is an attribute of matter, radiation, and non-gravitational
force fields. It is the source of the gravitational field in
the Einstein field equations of general relativity, just as
mass density is the source of such a field in Newtonian
gravity. The stress-energy-momentum tensor is symmetric.
Therefore, T^{μν} = T^{νμ}.
The covariant form of T is obtained from:
T_{μν} = T^{αβ}g_{αμ}g_{βν}
Continuity Equation
-------------------
The continuity equation describes the transport of a conserved
quantity. Continuity equations are the local form of conservation
laws. Basically, the total amount (of the conserved quantity)
inside any region can only change by the amount that passes in
or out of the region through the boundary. A conserved quantity
cannot increase or decrease, it can only move from place to
place.
First, consider the situation for charge.
Divergence corresponds to the flow. An outward flow corresponds
to a negative divergence. For a unit volume:
j^{μ} = (cρ,j^{x},j^{y},j^{z}) - the 4-current density.
∂ρ/∂t + ∇.j = 0
∂ρ/∂t + ∂j^{x}/∂x + ∂j^{y}/∂y + ∂j^{z}/∂z = 0
Write ∂ρ/∂t as ∂j^{0}/∂x^{o}. Therefore,
∂j^{μ}/∂x^{μ} = 0
By analogy with the conservation of charge we can also specify
an energy density and a momentum density and a flow of both in
and out of a small region. The momentum 4-vector is:
p^{μ} = (E/c,p^{1},p^{2},p^{3}).
Momentum flow would be the momentum passing through a unit area
per unit time which has the same units as pressure. Physically,
one can think of a gas at constant pressure in a box. If a hole
of unit area is opened in the side of the box, the pressure
would be the amount of momentum escaping per unit time.
Pressure = F/A where F = dp/dt = d(mv)/dt = ma.
= ma/A
[ma/A] = kg.(m/s^{2}).(1/m^{2})
= kg/ms^{2}
= 1 pascal
We can write a continuity equation for each row of T^{μν}:
∂T^{00}/∂t + ∂T^{01}/∂x + ∂T^{02}/∂y + ∂T^{03}/∂z = 0
or,
∂ρ_{e}/∂ct + ∇.T^{ti} = 0
Therefore, energy is conserved.
∂T^{10}/∂ct + ∂T^{11}/∂x + ∂T^{12}/∂y + ∂T^{13}/∂z = 0
Therefore, x-momentum is conserved. Likewise for the y
and z directions.
In shorthand:
D_{μ}T^{μν} = 0
Where D_{μ} is the covariant derivative.
Note: Conservation of energy-momentum only applies locally
(at an infinitesimal point). In large regions there is no
law. For example, redshifted light has a lower energy
than the source. This energy is lost - not conserved.
T_{μν} for a Perfect Fluid
-----------------------
Homogenious and isotropic spacetime can be considered to be
a perfect fluid. Perfect fluids have no shear stresses,
viscosity, or heat conduction. The Stress-Energy tensor of
a perfect fluid can be written in the form:
T_{μν} = (p + ρ_{e})U_{μ}U_{ν} + pg_{μν}
Where U is the 4-velocity vector field of the fluid. If
the fluid is only changing with time and not moving spacially
then U = (1,0,0,0). This gives:
- - - - - -
_{ } | p + ρ_{e} 0 0 0 | | -1 0 0 0 | | ρ_{e} 0 0 0 |
T_{μν} = | 0 _{ } 0 0 0 | + p| 0 1 0 0 | = | 0_{ } p 0 0 |
_{ } | 0 _{ } 0 0 0 | | 0 0 1 0 | | 0_{ } 0 p 0 |
_{ } | 0 _{ } 0 0 0 | | 0 0 0 1 | | 0_{ } 0 0 p |
- - - - - -
Where ρ_{e} = ρ_{m}c^{2}
T_{μν} for Dust
------------
A collection of particles not exerting pressure on each
other is often employed in cosmology as a model of a toy
universe, in which the dust particles are considered as
idealized models of galaxies, clusters, or superclusters.
T_{μν} = ρ_{e}U_{μ}U_{ν}
- -
_{ } | U_{t}U_{t} U_{t}U_{x} U_{t}U_{y} U_{t}U_{z} |
T_{μν} = ρ_{e}| U_{x}U_{t} U_{x}U_{x} U_{x}U_{y} U_{x}U_{z} |
_{ } | U_{y}U_{t} U_{y}U_{x} U_{y}U_{y} U_{y}U_{z} |
_{ } | U_{z}U_{t} U_{z}U_{x} U_{z}U_{y} U_{z}U_{z} |
- -
If the dust particles are only changing with time and not
moving spacially then U = (1,0,0,0). This gives:
- -
_{ } | ρ_{e} 0 0 0 |
T_{μν} = | 0^{ } 0 0 0 |
_{ } | 0^{ } 0 0 0 |
_{ } | 0^{ } 0 0 0 |
- -
Where again ρ_{e} = ρ_{m}c^{2}