If I had some line element such as minkowski line element:
ds2=−dt2+dx2+dy2+dz2
And assuming this photon or beam of photons travels in the x direction, how would one find the components of the tangent vector?
Actual answer should be kμ=B(1,1,0,0), but how is this obtained?
Answer
For an inertial observer with 4-velocity ta such that gabtatb=tbtb=−1,
let xa be the unit spacelike-vector orthogonal to ta.
So, gabxatb=xbtb=0 and gabxaxb=xbxb=1.
(In this frame, ta=(1,0,0,0) and xa=(0,1,0,0).)
So, ka=Ata+Bxa.
Since the photon tangent vector must satisfy gabkakb=kbkb=0,
we have:
0=kbkb=(Atb+Bxb)(Atb+Bxb)=A2tbtb+B2xbxb=−A2+B2
So, |A|=|B|.
Thus, for a future-directed, forward-pointing tangent vector, we have ka=Bta+Bxa, where B>0. In coordinates, this is ka=(B,B,0,0).
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