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Invariant Momentum

  1. Jul 14, 2006 #1
    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
     
  2. jcsd
  3. Jul 14, 2006 #2
    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
     
  4. Jul 14, 2006 #3

    pervect

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    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

    [tex]P^i = \int_V T^{i0} dV[/tex]

    where dV is an infinitesimal volume element expressed as a vector, and [itex]T^{ij}[/itex] 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

    [tex]\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[/tex]

    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 [itex]\frac{\partial}{\partial x^j}[/itex] with the covariant derivative [itex]\nabla_j[/itex]

    This gives the continuity equation in general coordinates as

    [tex]\nabla_a T^{ab} = 0[/tex]
     
    Last edited: Jul 14, 2006
  5. Jul 16, 2006 #4
    Thank you very much. These were quite helpful responses.
     
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