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Sum of null and time-like vectors

  1. Jan 1, 2016 #1
    1. The problem statement, all variables and given/known data

    Show that the sum of two future-pointing null vectors is a future-pointing time-like vector, except when the two null vectors have the same direction. Conversely, show that any time-like vector can be expressed as a sum of two null vectors. For a given time-like vector the two null vectors are not uniquely determined: what is the nature of the freedom in their choice?

    2. Relevant equations

    Under the 'mostly-minus' convention, a null vector ##N^{a}## satisfies ##N^{a}N_{a}=0##, a time-like vector ##T^{a}## satisfies ##T^{a}T_{a}>0##, and a space-like vector ##S^{a}## satisfies ##S^{a}S_{a}<0##.

    Given a future-pointing time-like vector ##P^{a}## (with ##P^{0}>0##), any other future-pointing time-like or null vector ##Q^{a}## satisfies ##P^{a}Q_{a}>0##.

    3. The attempt at a solution

    Consider two future-pointing null vectors ##P^{a}## and ##Q^{a}##. These vectors satisfy the following relations:

    ##P^{a}P_{a}=0,\ Q^{a}Q_{a}=0,\ P^{a}Q_{a}>0##.

    Therefore, ##(P^{a}+Q^{a})(P_{a}+Q_{a}) = P^{a}P_{a}+ 2\ P^{a}Q_{a} + Q^{a}Q_{a} = 2\ P^{a}Q_{a}##.

    Now, if ##P^{a} \not\propto Q^{a}##, then ##(P^{a}+Q^{a})(P_{a}+Q_{a}) = 2\ P^{a}Q_{a} > 0## and

    if ##P^{a} \propto Q^{a}##, then ##(P^{a}+Q^{a})(P_{a}+Q_{a}) \propto 2\ P^{a}P_{a} = 0##.

    Therefore, the sum of two future-pointing null vectors is a time-like vector, except when the two null vectors have the same direction, in which case the sum is again a null vector (regardless of the time-orientation of the original two vectors).


    Consider a time-like vector ##S^{a}+T^{a}##, where the nature of ##S^{a}## and ##T^{a}## are not yet known. The vector ##S^{a}+T^{a}## satisfies the following relation:

    ##(S^{a}+T^{a})(S_{a}+T_{a})>0##, which means that

    ##S^{a}S_{a}+T^{a}T_{a}+2S^{a}T_{a}>0##

    Where do I go from here?
     
  2. jcsd
  3. Jan 1, 2016 #2
    What happens if you now choose S and T to be null-like?
     
  4. Jan 1, 2016 #3
    If ##S^{a}## and ##T^{a}## are null-like, then ##S^{a}T_{a}>0##

    which means that ##S^{a}T_{a}## has to be future-pointing as well.


    But then, the relation ##S^{a}S_{a}+T^{a}T_{a}+2S^{a}T_{a}>0## is also satisfied for ##S^{a}## and ##T^{a}## future-pointing time-like?
     
  5. Jan 1, 2016 #4
    Yes, the question asks you to show that a time-like vector can be expressed as a sum of two null vectors; it did not say that that is the only way to decompose it. I don't see why you can't express a time-like vector as a sum of two time-like vectors if you wanted to.
     
  6. Jan 1, 2016 #5
    Thanks! Got it!

    Let me now answer the third part of the question:

    For a given time-like vector, the two null vectors are not uniquely determined: what is the nature of the freedom in their choice?

    The condition for the vectors ##T^{a}## and ##S^{a}## to be null is that ##T^{a}S_{a}>0##,

    so either both vectors are future-pointing, or both are past-pointing.

    Is this correct?
     
  7. Jan 1, 2016 #6
    Yup, that seems right to me.
     
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