J Baez' The meaning of Einstein's Equations

[DaTario]

J Baez' "The meaning of Einstein's Equations"

Hi All,

I am trying to study the paper of Jonh Baez in the july editon of Am. J. of Phys. on the meaning of Einstein's equation.

At a given point he says:

"The components of $T_{\alpha \beta}$ (stress-energy tensor) say how much momentum in the $\alpha$ direction is flowing in the $\beta$ direction through a given point of space-time, where $\alpha \beta$ = t, x, y, z. The flow of x-momentum in the x-direction is the pressure in the x-direction denoted $P_x$, and similarly for y and z. It takes a while to figure out why pressure is really the flow of momentum, but its eminently worth doing."

Does anybody know how to prove this last statement.


Thank you in advance



DaTario

 

[DrChinese]

You can find this at John's site: http://math.ucr.edu/home/baez/einstein/

 

[DaTario]

Thank you for the answer, but in his site there is the same information as in the paper I have in hands.

Best Regards

DaTario

 

[pervect]

Hi All,

I am trying to study the paper of Jonh Baez in the july editon of Am. J. of Phys. on the meaning of Einstein's equation.


DaTario

It's a bit hard to read your post since you haven't closed the latex tags properly. The latex support was added on to an existing system, so that [.itex] does not close with [.\itex] as one would expect, but rather with [./itex].
(Remove the dots!).

In answer to your question, imaginie a small box with a swarm of particles in it which move back in forth in only the x direction.

The particles will be transporting x momentum in both directions across the box. The stress energy tensor of this box will be represented by the energy density of the particles, plus an additional diagional component which represents the x- pressure.

If you are familar with how tensors transform, start with the stress-energy tensor of a fluid at rest, which is easy (it has only one term, the energy density T_00, or if you prefer T^00).

Then boost it so it consists of a fluid and/or swarm of particles moving in the +x direction.

hint: the transform is $$T^{cd} = T^{ab} L^c{}_a L^d{}_b$$

where $L^i{}_j$ is the transformation matrix, the same one that you would transform any 4-vector with, i.e.

[tex]
x^b = x^a L^b{}_a
[/itex]

[end hint]

Boost your original tensor again in the opposite direction, so it represents a fluid and/or swarm of particles moving in the -x direction.

Add the two boosted tensors together - you'll get a tensor that represents the situation I described, where particles move in both directions.

 

[DrChinese]

Thank you for the answer, but in his site there is the same information as in the paper I have in hands.

Best Regards

DaTario

I thought it might be a good idea to put it into the hands of the rest of us.

 

[DaTario]

Am. J. of Phys. july edition (printed version) 2005. pg 644.

Thank you for the explanation. It was really enlightening.

Best Regards,

DaTario