**************************
The energy-momentum tensor
**************************

.. include:: macros.rst
.. math:: \require{physics}

Dust, fluid, and flux
=====================

A fluid is a material which flows, that is, has forces perpendicular to
some imaginary surface which are much greater than the forces parallel
to it. A perfect fluid is the limit where the substance has pressure but
zero stress.

Dust is an idealised form of matter consisting of a collection of
non-interacting particles which are not moving relative to one-another,
and so the collection has zero pressure. Thus there is a momentarily
comoving reference frame (MCRF) with respect to which all of the
particles have zero velocity. If all of the particles have the same rest
mass, :math:`m`, but the cloud of dust may have a varying mass density,
:math:`n`.

Transforming to a frame movig at a velocity :math:`\vec{v}` to the MCRF
then the stationary volume element :math:`\Delta x \Delta y \Delta z`
will be Lorentz contracted to
:math:`\Delta x' \Delta y' \Delta z' = (\Delta x /
\gamma) \Delta y \Delta z`, for relative motion along the
:math:`x`-axis, increasing the number density to :math:`n \gamma`,
producing a flux through the area :math:`\Delta y \Delta z`. All the
particles pass through :math:`\Delta
y' \Delta z'` in a time :math:`\Delta t'` for
:math:`\Delta x' = v \Delta t'`, so the total number of particles is

.. math::
   
   (\gamma n) (v \Delta t') \Delta y' \Delta z'

This produces an :math:`x`-directed flux,

.. math::
   :label: flux
	   
   N^x = \gamma n v^x

or, defining a flux vector, :math:`\vec{N}`, and letting :math:`\vec{U} = (\gamma, \gamma v^x, \gamma v^y, \gamma v^z)` be the velocity 4-vector,

.. math::

   \label{eq:77}
     \vec{N} = n \vec{U}

In the MCRF :math:`\vec{U} = (1, \vec{0})`, so :math:`g(\vec{U}, \vec{U}) = -1` so

.. math:: g(\vec{N}, \vec{N}) = N_{\alpha} N^{\alpha} = -n^2

The components of the flux vector :math:`\vec{N}` in the frame are then

.. math:: \vec{N} = ( \gamma n, \gamma n \vec{v} )

Any function :math:`\phi(t,x,y,z)` over spacetime defines a constant surface, and its gradient :math:`\of{\dd}{\phi}` defines a normal to the surface.
The unit normal gradient is defined

.. math::

   \of{n} \equiv \frac{\of{\dd{}}{\phi}}{\abs{\of{\dd{}}{\phi}}}

Contracting this with the flux vector gives the flux across the corresponding surface.

The energy-momentum tensor
==========================

Energy and mass are interconvertible (special relativity); for a dust particle of mass :math:`m` the energy density of the dust (in the   MCRF) is :math:`E = mn`.
In a moving frame the number density becomes :math:`n \gamma`, and the energy of each particle is :math:`\gamma m`, giving a total energy of :math:`\gamma^2 mn` in a moving frame.
The :math:`\gamma^2` term cannot be generated by a simple Lorentz boost, so we require something of higher order.
Forming the :math:`(2,0)`-tensor

.. math::

     \label{eq:81}
       \ten{T} = \vec{p} \otimes \vec{N} = \rho \vec{U} \otimes \vec{U}

with :math:`\rho = mn` the mass density of the dust.
This is the energy-momentum (stress-energy) tensor.
The components of the tensor can be retrieved by contracting it with basis one-forms, :math:`\of{\omega}^{\alpha} = \of{\dd}x^{\alpha}`,

  .. math::
     :label: energymomentum-components
       \tensor{T}{^{\alpha \beta}} = \ten{T}(\of{\dd}x^{\alpha}, \of{\dd}x^{\beta}) = \vec{p}(\of{\dd}x^{\alpha}) \cp \vec{N}(\of{\dd}x^{\beta})

The :math:`\mathbf{00}` component, :math:`T^{00}` is the energy (flow of zeroth component of momentum across surface of constant time).
The :math:`\mathbf{0i}` component, :math:`T^{0i} = \gamma m \times \gamma n v^i` is the energy flux across a surface of constant :math:`x^i`.
The :math:`\mathbf{i0}` component, :math:`T^{i0} = p^i \times N^0 = m \gamma v^i \times \gamma n` is the flux of the :math:`i`\ th  component of momentum across a surface of constant time into the future.
This is the :math:`i`\ th component of momentum density.
This is the same as the energy flux, so

.. math::

   T^{i0} = T^{0i}

The :math:`\mathbf{ij}` component is the flux of :math:`i`-momentum across a surface of constant :math:`x^j`, and has the dimensions of a pressure.

In general :math:`\ten{T}` is symmetric, :math:`T^{\alpha \beta} = T^{\beta \alpha}`.

In a perfect fluid there is no preferred direction, so the spatial part of :math:`\ten{T}` is proportional to the spatial part of the metric, and there is no momentum transport perpendicular to the surface of a fluid element.
Thus

.. math::

   \label{eq:83}
     T^{ij} = p \delta^{ij}

for a perfect fluid.
Hence,

.. math::

   \label{eq:84}
     \ten{T} = (\rho + p) \vec{U} \otimes \vec{U} + p \ten{g}

Dust has no pressure, so in the MCRF it has

.. math::

   \label{eq:85}
     \ten{T} = \diag(\rho, 0, 0, 0)

The final property is the conservation law; if energy is conserved then the energy-momentum entering an arbitrary volume must be equal to that leaving it.
Thus

.. math::

   \label{eq:86}
     \pdv{x^0} T^{\alpha 0} + \pdv{x^1} T^{\alpha 1} + \pdv{x^2} T^{\alpha 2} + \pdv{x^3} T^{\alpha 3} = 0

That is

.. math::

   \label{eq:87}
     T^{\alpha \beta}{}_{, \beta} = 0

Similarly

.. math::

   \label{eq:88}
     N^{\alpha}{}_{, \alpha} = (n U^{\alpha})_{, \alpha} = 0