Energy-Momentum Equation of a Particle

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.

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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.

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

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

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

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

except you might need a delta function around the position.

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

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 consistant units anyway, to avoid confusion.

m2c4 = E2 - p2c2

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

ρm2c4 = ρE2 - ρp2c2

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...

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George Jones
Staff Emeritus