Conserving Lorentz Invariant Momentum in Particle Collisions

[MetaJoe]

Hi, All,

First time post, and this is quite possibly a very basic question: Is there a way to describe a particle's momentum such that the momentum itself is Lorentz invariant? The reason I am asking is this: As I understand it, if for example an electron and a positron were to collide and thus annihilate, such annihilation must (among other things) conserve momentum. What I'm looking for is a way to describe this momentum as it "carries though" the annihilation in such away that it is Lorentz invariant. Thank you so much!

MetaJoe

 

[pmb_phy]

Hi, All,

First time post, and this is quite possibly a very basic question: Is there a way to describe a particle's momentum such that the momentum itself is Lorentz invariant? The reason I am asking is this: As I understand it, if for example an electron and a positron were to collide and thus annihilate, such annihilation must (among other things) conserve momentum. What I'm looking for is a way to describe this momentum as it "carries though" the annihilation in such away that it is Lorentz invariant. Thank you so much!

MetaJoe

Hi Al and welcome to the forum.

Since the total 3-momentum of any closed system is conserved it follows that the total 4-momentum is also conserved. The magnitude of this 4-vector (the "invariant mass") is Lorentz invariant. For details please see

http://www.geocities.com/physics_world/sr/invariant_mass.htm

Pete

 

[pervect]

If an electron is at rest, it will have no momentum. If an electron is moving, it will have a non-zero momentum.

The difference between a moving electron and a stationary electron is just a Lorentz boost.

Therfore we do not expect the momentum of an electron to be Lorentz invariant - we expect it to change from zero when it is at rest to a non-zero value when we "boost" it.

The length of the enregy momentum 4-vector is an invariant as Pete says, however - it is the electron's rest mass.

In addition to the invariant rest mass, one can also write for a system a set of equations that represents the conservation of momentum, expressed in terms of the stress-energy tensor. These are known as the continuity equations.

The stress-energy tensor treats matter as a fluid, not as a collection of point particles. Therfore one sees laws that are similar to the laws of hydrodynamics, rather than laws written for a set of discrete particles.

The total momentum of a continuous system can be represented by

$$P^i = \int_V T^{i0} dV$$

where dV is an infinitesimal volume element expressed as a vector, and $T^{ij}$ is the stress-energy tensor.

A vector-valued volume element is just a 4-vector that is perpendicular to all spatial vectors in the volume element, and has a magnitude that's proportional to the volume.

The continuity equations

$$\frac{\partial T^{i0}}{d x^0} + \frac{\partial T^{i1}}{d x^1} + \frac{\partial T^{i2}}{d x^2} + \frac{\partial T^{i3}}{dx^3} = 0$$

can be regarded as a set of 4 equations (i=0,1,2,3) which represent the local conservation of energy and momentum.

The above equations are written for an orthonormal cartesian coordinate system. (Note that in such a cartesian coordinate system, the vector-valued volume element dV is just the time vector multipled by the volume element).

For an arbitrary coordinate system, one must replace the partial derivative $\frac{\partial}{\partial x^j}$ with the covariant derivative $\nabla_j$

This gives the continuity equation in general coordinates as

$$\nabla_a T^{ab} = 0$$

 

[MetaJoe]

Thank you very much. These were quite helpful responses.