Energy-momentum

[luc]

Right now I'm reading Hawking's "The large scale-structure of space time", and he uses constantly a quantity called Energy-momentum, though I can't see it defined anywhere. I've also read in some forum that in GR, total energy is not conserved, but energy-momentum is conserved. I think that perhaps energy-momentum is (Total energy + magnitude of 4-momentum), but I'm not sure. What exactly is energy-momentum?

[swansont]

There's an energy-momentum 4-vector, comprised of E and the three spatial components of pc. The scalar product with itself gives you E² - p²c², which is the square of the rest energy.

[luc]

'kay, but I think that 4-momentum is the same as the energy-momentum 4-vector (at least is Special Relativity is that way). So energy-momentum is the magnitude of the energy-momentum 4-vector, is that what you're saying?

[swansont]

'kay, but I think that 4-momentum is the same as the energy-momentum 4-vector (at least is Special Relativity is that way). So energy-momentum is the magnitude of the energy-momentum 4-vector, is that what you're saying?

 

I haven't read the book, so I can't say for sure.

 

IN GR there is also an energy-momentum tensor, aka the stress-energy tensor.

[□h=-16πT]

'kay, but I think that 4-momentum is the same as the energy-momentum 4-vector (at least is Special Relativity is that way).

 

The energy-momentum four-vector is the same as the 4-momentum vector. The 4-momentum vector has as its time component the energy of the frame and so gets its other name "energy-momentum" four-vector.

[Tom Mattson]

'kay' date=' but I think that 4-momentum is the same as the energy-momentum 4-vector (at least is Special Relativity is that way).

[/quote']

 

That's right.

 

So energy-momentum is the magnitude of the energy-momentum 4-vector, is that what you're saying?

 

That's not right. The norm of a particle's 4-momentum is the mass of the particle.

 

[math]

p^{\mu}=(E,\mathbf{p})

[/math]

[math]

p_{\mu}=(E,-\mathbf{p})

[/math]

 

Taking their inner product yields:

 

[math]

p^{\mu}p_{\mu}=E^2-p^2=m^2

[/math]

 

where [math]p[/math] is the norm of the 3-momentum.