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## Main Question or Discussion Point

The dominant energy condition is interpreted as meaning that no observer will see a flux of energy flowing at speeds greater than c. In this post I'm basically asking how to relate this concept of "speed of the energy flux" more precisely to the stress-energy tensor.

If we start with dust, described in its rest frame, we have a stress-energy tensor that looks like this:

[tex]

\left(\begin{matrix}

1 & 0 \\

0 & 0

\end{matrix}\right)

[/tex]

This is in coordinates (t,x), ignoring an over-all multiplicative constant.

Boosting along x gives a stress-energy tensor that looks like this:

[tex]

\left(\begin{matrix}

1 & v \\

v & v^2

\end{matrix}\right)

[/tex]

This is exact, not approximate, although I'm leaving out a factor of [itex]\gamma^2[/itex] in front, since I don't care about multiplicative constants.

The DEC says that if I multiply this matrix by a vector that lies in or on the future light-cone, I get another vector that lies in or on the cone...but this doesn't seem to me to connect in any obvious way to the concept of a flux of mass-energy flowing with some velocity.

If I let [itex]v=1-\epsilon[/itex], with [itex]|\epsilon| \ll 1[/itex], then [itex]\epsilon=0[/itex] gives the correct stress-energy tensor for, e.g., an electromagnetic wave, and [itex]\epsilon<0[/itex] gives one that violates the DEC.

In the above example, [itex]T_{xt}/T_{tt}[/itex] is the velocity at which mass-energy flows in the x-t frame. However, this doesn't seem to be a valid interpretation in general. For example, if I start with

[tex]

\left(\begin{matrix}

1 & p \\

p & q

\end{matrix}\right) ,

[/tex]

and I want to find a velocity v such that a boost eliminates the off-diagonal component, then I only get [itex]v \approx -q[/itex] in the case where both p and q are small.

In terms of old-fashioned E&M, the existence of a nonzero Poynting vector doesn't indicate that there is a flow of energy; that requires a nonzero divergence for the Poynting vector. By analogy, it seems unlikely to me that one could conclude the "speed of energy flux" from knowing T at a point, without knowing T's derivatives. If I put together a static EM field with a nonzero Poynting vector, then I get a stress-energy tensor where basically [itex]T_{tt}[/itex] is the energy density and [itex]T_{tx}[/itex] is the Poynting vector. (See http://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor .) So clearly the ratio [itex]T_{xt}/T_{tt}[/itex] doesn't tell me the "speed of the energy flux."

So it seems that it is not possible at all to read off the "speed of the energy flux" from T. In that case, what is the justification for this interpretation of the DEC? The example above with the epsilons makes it plausible to me, but it only seems like a plausibility argument.

If we start with dust, described in its rest frame, we have a stress-energy tensor that looks like this:

[tex]

\left(\begin{matrix}

1 & 0 \\

0 & 0

\end{matrix}\right)

[/tex]

This is in coordinates (t,x), ignoring an over-all multiplicative constant.

Boosting along x gives a stress-energy tensor that looks like this:

[tex]

\left(\begin{matrix}

1 & v \\

v & v^2

\end{matrix}\right)

[/tex]

This is exact, not approximate, although I'm leaving out a factor of [itex]\gamma^2[/itex] in front, since I don't care about multiplicative constants.

The DEC says that if I multiply this matrix by a vector that lies in or on the future light-cone, I get another vector that lies in or on the cone...but this doesn't seem to me to connect in any obvious way to the concept of a flux of mass-energy flowing with some velocity.

If I let [itex]v=1-\epsilon[/itex], with [itex]|\epsilon| \ll 1[/itex], then [itex]\epsilon=0[/itex] gives the correct stress-energy tensor for, e.g., an electromagnetic wave, and [itex]\epsilon<0[/itex] gives one that violates the DEC.

In the above example, [itex]T_{xt}/T_{tt}[/itex] is the velocity at which mass-energy flows in the x-t frame. However, this doesn't seem to be a valid interpretation in general. For example, if I start with

[tex]

\left(\begin{matrix}

1 & p \\

p & q

\end{matrix}\right) ,

[/tex]

and I want to find a velocity v such that a boost eliminates the off-diagonal component, then I only get [itex]v \approx -q[/itex] in the case where both p and q are small.

In terms of old-fashioned E&M, the existence of a nonzero Poynting vector doesn't indicate that there is a flow of energy; that requires a nonzero divergence for the Poynting vector. By analogy, it seems unlikely to me that one could conclude the "speed of energy flux" from knowing T at a point, without knowing T's derivatives. If I put together a static EM field with a nonzero Poynting vector, then I get a stress-energy tensor where basically [itex]T_{tt}[/itex] is the energy density and [itex]T_{tx}[/itex] is the Poynting vector. (See http://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor .) So clearly the ratio [itex]T_{xt}/T_{tt}[/itex] doesn't tell me the "speed of the energy flux."

So it seems that it is not possible at all to read off the "speed of the energy flux" from T. In that case, what is the justification for this interpretation of the DEC? The example above with the epsilons makes it plausible to me, but it only seems like a plausibility argument.

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