# Inertial mass of a stressed body

pmb
I was wondering if anyone out there knows how to do Exercise 5.4 on page 159 of "Gravitation," by Misner, Thorne and Wheeler? It goes like this - let d^jk = Kronecker delta

Consider a stressed medium in motion with ordinary velocity |v|<<1 with respect to a specific Lorentz frame

(a) Show by Lorentz transformations that the spatial components of the momentum density as

(5.51) T^0j = Sum(k) m^jk v^k,

where

(5.52) m^jk = T^0'0' d^ji + Tj'k'

and T^u'v'are the components of the stress-energy tensor in the rest frame of the medium

(b) Derive equations (5.51) and (5.52) from Newtonian considerations plus the equivalence of mass-energy. (Hint; the total mass-energy carried past the observer by a volume V in the medium includes both rest mass T^0'0'V and the work done by forces acting across the volume faces as they "push" the volume through a distance.)

(c) As a result of relation (5.51), the force per unit volume required to produce an acceleration dv^k/dt in a stressed medium, which is at rest with respect to the man who applies the force, is

(5.53) F^j = dT^0j/dt = Sum(k) m^jk dv^k/dt

This equation suggests that one call m^jk the "inertial mass per unit volume" of a stressed medium at rest. In general m^jk is a symetric 3-tensor. What does it become for the special case of a perfect fluid?

(d) Consider an isolated, stressed body at rest and in equilibrium (T^ab,0 = 0) in the laboratory frame. Show that its total inertial mass, defined by

M^jk = Integral(over stressed body) m^jk dx dy dz

is istropic and equals the rest mass of the body

M^jk = d^jk * Integral(over stressed body) T^00 dx dy dz

This is not a homework problem per se. I'm doing a self-study and don't see how MTW get part (a). So any hints on how to get to (a) will be appreciated - we'll go from there. I tried a boost in an arbitrary direction (see page 69 of MTW) but it doesn't seenm to work.

This does raise an interesting question - What would it mean to call T^uv/c^2 the "mass tensor"?

Pete

jeff
Originally posted by pmb
Consider a stressed medium in motion with ordinary velocity |v|<<1 with respect to a specific Lorentz frame

(a) Show by Lorentz transformations that the spatial components of the momentum density as

(5.51) T^0j = Sum(k) m^jk v^k,

where

(5.52) m^jk = T^0'0' d^ji + Tj'k'

and T^u'v'are the components of the stress-energy tensor in the rest frame of the medium I'm doing a self-study and don't see how MTW get part (a). So any hints on how to get to (a) will be appreciated - we'll go from there. I tried a boost in an arbitrary direction (see page 69 of MTW) but it doesn't seenm to work.

Neglecting terms of 2nd order in velocity, the lorentz transformation is

&Lambda;00&prime; = 1 , &Lambda;0i&prime; = vi , &Lambda;ij&prime; = &delta;ij&prime;

If Tu&prime;v&prime; is the stress-energy-momentum tensor in the fluid's rest frame, then T0&prime;i&prime; = 0.

Now compute

T0j = &Lambda;0u&prime;&Lambda;jv&prime;Tu&prime;v&prime;

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DavidW
What I think is preventing you from being able to do the problem is that you are rigidly stuck in your awkward definitions of mass etc. For a better version of the problem see problem 3.1.4 of "Modern Relativity" at

http://www.geocities.com/zcphysicsms/chap3.htm#BM26

As was explained to you by others and included in a FAQ,

"And this is another reason why, in the end, it's so much easier to just take the mass to be the invariant quantity m, and to put any directional information into a separate, matrix, factor."

http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

pmb
What I think is preventing you from being able to do the problem ...

Oy! Total nonsense. I just didn't see the expansion do to a silly oversight on my part. jeff was kind enough to clear that up for me - End of story! And please stop repeating yourself over and over and over and over and over and over and over and over and over and again

Folks - This person follows me from forum to forum repeating himself to the point of nausea. He has a *very* limited understanding of relativistic mass and passes around misinformation. I don't have the time to correct him here as well as everywhere else.

Pmb

Does stress increase the inertia of a body? Does it increase gravitational attraction or in any way affect spacetime curvature? If so, I'm confused, I thought it took energy to do these things.

pmb
Originally posted by Sacroiliac
Does stress increase the inertia of a body? Does it increase gravitational attraction or in any way affect spacetime curvature? If so, I'm confused, I thought it took energy to do these things.

Yes. Stress increases the inertia of a body as well as it's ability to generate a gravitational field. Energy in one frame is stress in another frame - this implies that even though a body is at rest the increased stress increases the inertia and the g-field in creates.

It's for this reason that pressure is a source of gravity.

Pete

Originally posted by pmb
Yes. Stress increases the inertia of a body as well as it's ability to generate a gravitational field. Energy in one frame is stress in another frame - this implies that even though a body is at rest the increased stress increases the inertia and the g-field in creates.

It's for this reason that pressure is a source of gravity.

Pete

Your's was the clearest answer I've ever gotten to that question, but could you possiblly give me some kind of an example of "Energy in one frame is stress in another frame."

And when you say "It's for this reason that pressure is a source of gravity.", does this mean that stress is the result of pressure? I've always been confused about whether both stress and pressure are sources of gravity or whether one is the cause of the other.

Sorry that last post was by Sacroiliac not Ring. He and I post from the same computer and he won't be happy about this.

pmb
Originally posted by Ring
Your's was the clearest answer I've ever gotten to that question, but could you possiblly give me some kind of an example of "Energy in one frame is stress in another frame."

And when you say "It's for this reason that pressure is a source of gravity.", does this mean that stress is the result of pressure? I've always been confused about whether both stress and pressure are sources of gravity or whether one is the cause of the other.

Well it's nice to hear that I'm clear to at least some people! :-)

It's a lot easier to say in mathematics. But consider an EM analogy. Suppose there is just a charged particle sitting -- at rest -- in your frame of referance. Then all you need to do is to specify the magnitude of the charge and where it is and then you pretty much have all you need in order to describe the field right? Now suppose that the charge is moving! Now if you want to descibe the field you need more then the charge - you need the velocity of the charge. What is a moving charge? It's current right?

If you have a mathematical quantity which describes the EM sources then it must have a component which describes the charge and one component which describes the current. You can describe this completely with two functions: Charge density and current density. In special relativity there is a mathematical object which is used to model this information - it's called the 4-current. One component is charge density. The other three components are the componets of a spatial 3-vector the current density. Let J^u be the 4-current. I'll represent this as follows

J^u = (J^0, J^1, J^2, J^3)

J^0 is the charge density
J^k = current density

Now I can change from one inertial to another with a Lorentz transformation. In one frame, the non-primed frame, I'll label it as J^u - I can represent it with just one symbol - J. In a fame moving relative to the first, the primed frame, I'll label it as J'^v. I can represent this with one symbols too - J'

They're related by a Lorentz tranformation as follows

J' = LJ

where this is matrix multiplication. In the non-primed frame I can have

J = (rho, 0, 0, 0) ---> no current in this frame

in the primed frame I can have

J' = (rho', J'^1, J'^2, J'^3) ---> there *is* current in this frame

So I can say that charge in one frame is charge and current in another frame.

Same with the above with mass/energy - just more complicated.

I'll try to elaborate more later - at the moment I ahve sciatica and am in extreme pain - the medication takes care of the pain but makes my kind of "loopy"

Pete

DavidW
Originally posted by pmb
Yes. Stress increases the inertia of a body as well as it's ability to generate a gravitational field. Energy in one frame is stress in another frame - this implies that even though a body is at rest the increased stress increases the inertia and the g-field in creates.

It's for this reason that pressure is a source of gravity.

Pete

pmb is confused as always. Anything that goes into the stress-energy tensor is a source in Einstein's field equations of gravity. What is wrong is the statement that energy in one frame is stress in another. For counter example start with a hypothetical sphere of mass under no stress or pressure according to the center of momentum frame. The only nonzero term of the stress energy tensor according to that frame is the energy density term T^00. A frame transformation for a simple boost results in a stress energy tensor with both energy density and momentum density terms T^0i that are not zero but does no meaningful transformation results in nonzero stress terms T^ij. You'd have to pick some very meaningless screwy coordinates to pick up those terms.

pmb
Originally posted by DavidW
pmb is confused as always. Anything that goes into the stress-energy tensor is a source in Einstein's field equations of gravity. What is wrong is the statement that energy in one frame is stress in another. For counter example start with a hypothetical sphere of mass under no stress or pressure according to the center of momentum frame. The only nonzero term of the stress energy tensor according to that frame is the energy density term T^00. A frame transformation for a simple boost results in a stress energy tensor with both energy density and momentum density terms T^0i that are not zero but does no meaningful transformation results in nonzero stress terms T^ij. You'd have to pick some very meaningless screwy coordinates to pick up those terms.

sparky here misses the point as usual. That was meant as a general comment regarding the components of the stress-energy-momentum tensor. There are situations where there is no stress in all frames as sparky snips about = But there are examples where it is as well.

However sparky have never been one who's interested in discussion - he's only seeking new new ways to irritate and insult.

sparky - Someone is confused when they're told what a scalar is, given an enormous amnount of proof, then sticks to an incorrect position for the sole reason that he can't admit to making an error.

Find one place to flame me and stick to that. Don't follow me around to to it.

pmb
Originally posted by Sacroiliac
Sorry that last post was by Sacroiliac not Ring. He and I post from the same computer and he won't be happy about this.

If you have any further questions please e-mail them to me at [email protected]

This waite charater is using this thread to be irritating and harrassing - i.e. He follows me from forum to forum interupting conversations that I get into for the sole purpose of causing trouble. This thread will be deleted tomorrow.

Pmb