Energy-Momentum Equation of a Particle

[Phrak]

How do I get from the Energy-Momentum equation of a particle to its Stress-Energy equation?

By way of introducing the energy-momentum equation:

For a single particle, in units where c=1, a relationship between mass, energy and momentum appear as a direct result of the 4-velocity:

$$m^2 = E^2 - p^2$$

We can define a 4-vector,

$$m^\mu = (E^0, p^i) \ ,$$

(Greek indices ranges from 0 to 3, Latin indices ranges from 1 to 3 indexing spatial coordinates, and 0 is the temporal coordinate.)

Mass is defined as the magnitude of the 4-vector,

$$m^2 = m^\mu m_\mu \ .$$​

If the world line of the particle in question passes through some 4-volume, equally distrubuted in the volume, dtdxdydz we should have a stress-energy energy equation for this volume. Or am I wrong?

...I guess it's better to ask about an infinitessimal mass, dm, or about a finite 4-volume delta t delta x delta y delta z. Either way, it would be wonderful to know how to approach this problem.

 

[Phrak]

My notation may be confusing. I should use different symbols for intrinsic mass and the mass 4-vector, and make a few other touch-ups.

$$m^2 = E^2 - \textbf{p}^2$$

$$\hat{m}^\mu = (E, p^i)$$

$$m^2 = \hat{m}^\mu \hat{m}_\mu$$

 

[Mentz114]

What about

$$T^{\mu\nu}=\rho u^{\mu} u^{\nu}$$

except you might need a delta function around the position.

 

[Phrak]

What about

$$T^{\mu\nu}=\rho u^{\mu} u^{\nu}$$

except you might need a delta function around the position.

I think you're on the right track. Bringing in the 4-velocity was a good idea. In mulling this over, this afternoon, a good place to start is with the time-time component of the stress-energy tensor.

In common units, T⁰⁰ has units of relativistic mass per unit volume. We want energy units, but we can get energy density by multiplying by c².

The equation m² = E² - p² can be taken as an equation about mass, energy and momentum densities, instead, by dividing each term by a unit volume. It's not quite solving for a particle moving in some trajectory, but similar.

Come to think of it, m² = E² - p², should be put in consistent units anyway, to avoid confusion.

m²c⁴ = E² - p²c²

So now it's in energy units which is what we wanted. In terms of spatial densities,

ρ_m²c⁴ = ρ_E² - ρ_p²c²

At this point we have to be careful, because it's not Lorentz invariant. I divided by a volume element which is not Lorentz invariant...

 

[George Jones]

Take a look at

https://www.physicsforums.com/showthread.php?p=1389549#post1389549.

 

[Phrak]

George. You’ve changed your picture. Nice little family. Best of times to you.

I’m trying to find some deeper meaning to the mass-energy-momentum equation. The heuristic, m² = E² - p², says nothing about the underlying continuum that gives rise to the particle, yet there it is. It also seems far too linear to arise from a nonlinear theory of spacetime, except as a low energy solution.

The heuristic is to go back to Minkowski space, or an asymptotically flat part of spacetime, invoke the 4-velocity as a direct consequence of relativity and claim the 4-velocity, rescaled by mass, is a local property of a body at some given (t,x,y,z).