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Tensor Equation

  1. Apr 18, 2006 #1
    If a tensor equation is true in all reference frames and vectors are tensors of rank 1, then why aren't equations involcing velocity or momentum true in all reference frames?
    why do they have to be revised, as, say, in special relativity ( the lorentz transformations)?


    thanks
     
  2. jcsd
  3. Apr 18, 2006 #2

    Garth

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    If a vector equation is true in one reference frame then it is true in all inertial reference frames.

    A + B = C, where A, B and C are all vectors, is a frame independent statement.

    What is not frame independent is the coordinate representation of these vectors in a certain vector base. The Lorentz transformations show how these bases transform.

    Garth
     
    Last edited: Apr 18, 2006
  4. Apr 18, 2006 #3

    HallsofIvy

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    "True in all reference frames" means that if vector a= vector b in one reference frame then vector a= vector b in all reference frames. It does not mean that vector a in one reference frame = vector a in another reference frame which is what you appear to be thinking about when you refer to Lorentz transformations.
     
  5. Apr 18, 2006 #4

    Meir Achuz

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    It is a 4-vector equation that is true in all Lorentz frames.
    You seem to be referring to 3-vectors.
    Equations involving 4-velocity or 4-momentum are true in all Lorentz frames
     
  6. Apr 19, 2006 #5

    Garth

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    As we are dealing with Lorentz transformations then the vectors we are dealing with are 4-vectors, we are dealing with objects in both space and time.

    A 3-vector is the projection of a 4-vector onto a particular foliation of space-time in a particular frame of reference.

    These 3-vector components of a 4-vector are dependent on the particular 'slice' or foliation of space-time, which is dependent on the particular frame of reference.

    Lorentz transformations convert from one particular foliation to another, that is from one observer to a second observer moving relative to the first.

    Garth
     
    Last edited: Apr 19, 2006
  7. Apr 19, 2006 #6
    velocity (= 3-velocity, i.e. the regular velocity that you know and love) and momentum (=3-momentum, i.e. regular momentum) are vectors in 3-dimensional space[i.e. the sum of all (x,y,z)]. I non-relativistic mechanics a (covariant) vector is anything whose components transform in the same way as the position vector r under an orthogonal transformation (i.e. rotations of axes) if and only if the space is flat (unlike the Earth's surface). In this way you can add 3-vectors with no problem. In relativity one speaks of 4-velocity and 4-momentum which are objects which lie in a 4-dimensional space aka spacetime [i.e. the sum of all (t, x, y, z)]. Thus if and only if the 4-dimensional space is flat you can add these vectors in a unique way. If the spacetime is curved then the resulting sum is not unique and will depend on how you "transport" one vector to the other which may be located at different places.

    Pete
     
  8. Apr 21, 2006 #7
    Thanks all for the great replies!

    blumfeld0
     
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