Energy-Momentum Equation of a Particle

In summary, the energy-momentum equation can be used to calculate the mass, energy and momentum densities for a particle in a given (t,x,y,z) space. The equation is not Lorentz invariant, so care must be taken when working with it.
  • #1
Phrak
4,267
6
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:

[tex]m^2 = E^2 - p^2[/tex]

We can define a 4-vector,

[tex]m^\mu = (E^0, p^i) \ ,[/tex]

(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,

[tex]m^2 = m^\mu m_\mu \ .[/tex]​

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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  • #2
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.

[tex]m^2 = E^2 - \textbf{p}^2[/tex]

[tex]\hat{m}^\mu = (E, p^i) [/tex]

[tex]m^2 = \hat{m}^\mu \hat{m}_\mu[/tex]
 
  • #3
What about

[tex]
T^{\mu\nu}=\rho u^{\mu} u^{\nu}
[/tex]

except you might need a delta function around the position.
 
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  • #4
Mentz114 said:
What about

[tex]
T^{\mu\nu}=\rho u^{\mu} u^{\nu}
[/tex]

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 spatial 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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  • #6
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).
 
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1. What is the Energy-Momentum Equation of a Particle?

The Energy-Momentum Equation of a Particle is a fundamental equation in physics that describes the relationship between a particle's energy and momentum. It states that the total energy of a particle is equal to the sum of its kinetic energy and potential energy, while its momentum is equal to its mass multiplied by its velocity.

2. How is the Energy-Momentum Equation of a Particle derived?

The Energy-Momentum Equation of a Particle is derived from the laws of conservation of energy and momentum, as well as the principles of classical mechanics. It is based on the concept of work, which is defined as a force acting on an object over a distance. By applying these principles, the equation can be derived and used to analyze the behavior of particles in various physical systems.

3. What is the significance of the Energy-Momentum Equation of a Particle?

The Energy-Momentum Equation of a Particle is significant because it allows us to understand and predict the behavior of particles in different physical systems. It is used in various fields of physics, including mechanics, thermodynamics, and electromagnetism. It also forms the basis for more complex equations, such as the relativistic energy-momentum equation, which is used to describe the behavior of particles at high speeds.

4. How is the Energy-Momentum Equation of a Particle related to Newton's laws of motion?

The Energy-Momentum Equation of a Particle is closely related to Newton's laws of motion. In fact, it can be derived from Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. By combining this law with the concept of work, we arrive at the Energy-Momentum Equation of a Particle, which provides a more comprehensive understanding of a particle's behavior.

5. Can the Energy-Momentum Equation of a Particle be applied to all types of particles?

Yes, the Energy-Momentum Equation of a Particle can be applied to all types of particles, including those with mass and those without mass (such as photons). However, it is important to note that for particles with mass, the equation is based on classical mechanics and may not accurately describe the behavior of particles at high speeds. In these cases, the relativistic energy-momentum equation should be used instead.

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