My problem is how to calculate the radiation measured from a moving star. I figure there's two ways to do it, both of which I go through below, but they do not produce the same answer!(adsbygoogle = window.adsbygoogle || []).push({});

For the first part everything is in the rest frame, [itex] \mathcal{O} [/itex], of the star.

The star emits [itex] N [/itex] photons of frequency [itex] \nu [/itex] each second. The total energy radiated per second is thus [itex] L := Nh\nu [/itex]. It can be shown the stress-energy tensor in this case has entries for events [itex] (t,x,0,0) [/itex]:

[tex] T^{00} = T^{0x} = T^{x0} = T^{xx} = \frac{L}{4\pi x^2}, [/tex]

and all other entries zero.

We now move to the frame of the observer, [itex] \overline{\mathcal{O}} [/itex], who moves with speed [itex] v [/itex] in the positive [itex] x [/itex]-direction and who is also located on the x-axis. The components of the stress-energy tensor in this frame are given by (using the usual rules for tensor transformation):

[tex] T^{\overline{\alpha} \overline{\beta} } = \Lambda^{\overline{\alpha}}_{\mu} \Lambda^{\overline{\beta}}_{\nu} T^{\mu \nu} [/tex]

Now if the coordinate of reception is [itex] x [/itex] in the star's frame it must be [itex] R = (1-v)\gamma x [/itex] in the observer's frame (with [itex] \gamma = (1-v^2)^{-1/2} [/itex]). Thus we have that

[tex] T^{\overline{0} \overline{x}} = \frac{L}{4\pi x^2} (1-v)^2 \gamma^2, [/tex]

[tex] T^{\overline{0} \overline{x}} = \frac{L}{4\pi R^2} (1-v)^4 \gamma^4. [/tex]

So this gives the radiation the observer measures. However if we approach the problem using doppler-shift of the photons we get [itex] \nu' = (1-v)\gamma \nu [/itex], [itex] R = (1-v)\gamma x [/itex], and we have

[tex] T^{\overline{0} \overline{x}}= \frac{Nh\nu}{4\pi R^2} (1-v)^3 \gamma^3. [/tex]

So which is correct? I'm fairly sure the first method is correct, but I can't figure out exactly what I'm missing in the second calculation.

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# Radiation measured from moving star

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