- #1

pmb

## Main Question or Discussion Point

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"?

Thank you in advance

Pete

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"?

Thank you in advance

Pete