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SR from GR

  1. Jul 8, 2007 #1

    quasar987

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    Does SR appear as the special case of GR when [itex]T_{\mu\nu}=0[/itex] in which case the solution of Einstein's equation is the Minkowski metric?

    And what are the Ricci tensors and scalar curvature like in the case [itex]T_{\mu\nu}=0[/itex]?
     
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  3. Jul 8, 2007 #2

    robphy

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    Using the field equations (with zero cosmological constant), a zero stress tensor yields a zero ricci tensor and ricci scalar. You'll need more than requiring "vacuum" (zero stress tensor) to get SR.

    You'll need a zero Weyl Tensor as well.
    Strictly speaking, to get "SR", you'll need the right manifold, [tex]R^4[/tex], to start with.
     
  4. Jul 8, 2007 #3

    pervect

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    The analogy to Maxwell's equations might be helpful here to clarify some of the reasons why a zero stress tensor doesn't guarantee a Minkowski (SR) metric.

    Consider asking "Suppose you have no charges - are the E and B fields zero everywhere?". The answer is no, you could have electromagnetic radiation. Usually one specifies boundary conditions as well as a charge distribution to get a unique solution to Maxwell's equations. For Maxwell's equations, having E and B zero at infinity is a standard boundary condition, for GR the analogous boundary condition would be "asymptotic flatness".
     
  5. Jul 8, 2007 #4

    robphy

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    Are you suggesting that, for GR, vacuum and asymptotic flatness imply that the Riemann curvature is zero everywhere?
     
    Last edited by a moderator: Jul 9, 2007
  6. Jul 8, 2007 #5
    SR is the special case of GR where the metric components [itex]g_{\mu\nu}[/itex] are all constants. This corresponds to a zero Riemann tensor, which is more restrictive than setting the Ricci tensor equal to zero. [itex]T_{\mu\nu}=0[/itex], which gives a zero Ricci tensor, corresponds to empty space. Empty space can be curved, of course.
     
  7. Jul 9, 2007 #6

    pervect

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    No, not really.
     
  8. Jul 9, 2007 #7
    I've used [itex]T_{\mu\nu}[/itex] in SR myself. In fact I learned more about it in an SR text than any GR text that I have. In my cases it was used to analyze things like the non-proportionality of energy density and inertial mass density. As far as [itex]T_{\mu\nu}= 0[/itex], this can lead to the Minkowski metric once you transform to an inertial frame. Otherwise the components of the metric (i.e. the set of ten gravitational potentials g_uv) may net be constant in space, i.e. there could be gravitational forces/accelerations still present.

    Pete
     
    Last edited: Jul 9, 2007
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