Stress energy tensor for a swarm of particles ( in MTW )

[zn52]

hey Folks,
Please see attachment. I'm in doubt about the equation 5.15a. Indeed, it is said on the line just above it that : "in a frame where particles have velocity Va" which means the lab frame , say. Then in this frame the time component of 4-velocity is the Lorentz contraction factor whereas here it is taken as 1 ? But here the factor is due to the volume contraction ... Can someone clarify please ?
Thanks,

 

[pervect]

hey Folks,
Please see attachment. I'm in doubt about the equation 5.15a. Indeed, it is said on the line just above it that : "in a frame where particles have velocity Va" which means the lab frame , say. Then in this frame the time component of 4-velocity is the Lorentz contraction factor whereas here it is taken as 1 ? But here the factor is due to the volume contraction ... Can someone clarify please ?
Thanks,

MTW is saying that the number density in the lab frame is the number density in the particle's frame, which is moving relative to the lab, multiplied by the contraction factor.

You are supposed to imagine all particles all having all components of their velocities their same, such that any two particles have a constant distance between them.

If you look at a pair of such particles, said distance will be contracted by some factor gamma in the lab frame along the axis of motion, compared to the distance in the particle's frame, because of the standard Lorentz distance contraction.

There won't be any distance contraction in the directions perpendicular to the axis of motion.

Hope that helps?

 

[zn52]

I agree . But I still have some nagging points which I would appreciate if you could clarify them to me and the reader , please correct me if I'm wrong:

1 - You mentioned that nbr density in lab frame = nbr density in particles' frame * Gamma

This means : n/v = (N/V) * Gamma where n is the number of particles in the particles' frame and N is the number of particles in the lab frame and Gamma is 1/sqrt(1-vv) .

V = v / Gamma and if we plug this in the equation above we would get n(1-vv) = N ?
but n = N !

2 - How can we prove that V = v / Gamma ?

Thank you for your help,
my best regards,

 

[pervect]

Draw a space-time diagram, and mark off the "volume element" (it'll only be a line segment with a 2-d space-time diagram) in the lab frame, and the "volume element" in the moving frame.

Note that they have different lengths, and different orientations.

I assume that this volume element is what you're calling V above, it wasn't clear. The fact that V refers to a different set of points in the particle frame and in the lab frame is what's important to know.

Because different observers have different notions of what a unit volume element is, one needs some machinery to compute the density for an arbitary observer. This machinery is just the stress-energy tensor, you feed the stress energy tensor the four-velocity, and it spits out the density, which depends on the observer.

 

[zn52]

I thank you so much for your clarification. Indeed I would have thought that the volume represents the same set of points but I was wrong. I had done an excercise in the book : "Problem book in relativity and Gravitation" from which I have attached one solution.

As you can see volumes are related via the Jacobian determinant which as far as I can tell is Gamma squared for the matrix attached. Does that mean that V' = V / Gamma squared , I mean
do we then have : V' = V / (1-vv)
This would then mean that volume contraction is the square of the length contraction ?

Doesn't that make sense ? I'm so sorry but I'm really confused with this

PS : Matrix taken from : http://en.wikipedia.org/wiki/Lorentz_transformation

PS : the book is shown here : https://www.amazon.com/dp/069108162X/?tag=pfamazon01-20

 

[zn52]

Oops look what we have. It seems since dv' = Jacobian * dv and since dv = dtdxdydz and because dt and dt' are related by Gamma which means that dx'dy'dz' = Gamma * dxdydz

Yuppy ! now I'm convinced and can sleep at night happily. By the way I have just seen that the chapter in Schutz' Intro to GR is very well done as to the energy Momentum Tensor...it also contains stuff related to number density and whatnot...

Thank you very much for your assistance sir and sorry for my stupidness since I'm a bit stubborn and do not leave anything which I do not understand...

with my best regards from the south of France.