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Ibix

Science Advisor

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I was looking at null geodesics in Schwarzschild spacetime. Carroll's lecture notes cover them here: https://preposterousuniverse.com/wp-content/uploads/grnotes-seven.pdf

He notes (and justifies) that orbits lie in a plane and chooses coordinates so it's the equatorial plane, then uses Killing vectors to identify conserved quantities and put the geodesic equations in an effective potential form:$$\begin {eqnarray}\frac 12 E^2&=&\frac 12\left (\frac {dr}{d\lambda}\right)^2+V (r)\\

V (r)&=&\frac {L^2}{2r^2}-\frac {GML^2}{r^3} \end {eqnarray}$$(Carroll's 7.47 & 7.48, where I've set ##\epsilon=0## for a null geodesic). Carroll says that E and L are the energy and angular momentum of the light pulse and says (his equations 7.43 & 7.44)$$\begin{eqnarray}E&=&\left (1-\frac {2GM}r\right)\frac {dt}{d\lambda}\\

L&=&r^2\frac {d\phi}{d\lambda}\end {eqnarray} $$

My question is: if I am hovering at r armed with a protractor and a laser pointer, how do I relate my protractor reading to L and E in order to determine where my beam goes?

I can rearrange (1) to get $$\frac {dr}{d\lambda}=E\sqrt {1-\frac {L^2}{E^2r^2}\left (1-\frac {2GM}r\right)} $$which tells me that E isn't important on its own - it's a scale factor, but since ##\lambda## has no particular physical significance it comes out in the wash. What is important is L/E. Dividing (4) by (3), then, I think I can write $$\frac LE =\frac r {1-2GM/r}\left (r\frac {d\phi}{dt}\right) $$The term in brackets is just the locally measured tangential component of the "muzzle velocity" of my beam. So I can replace it (in the c=1 units) with ##\cos\psi##, where ##\psi## is my protractor reading (zero points to the center of the black hole).

Is that right?

He notes (and justifies) that orbits lie in a plane and chooses coordinates so it's the equatorial plane, then uses Killing vectors to identify conserved quantities and put the geodesic equations in an effective potential form:$$\begin {eqnarray}\frac 12 E^2&=&\frac 12\left (\frac {dr}{d\lambda}\right)^2+V (r)\\

V (r)&=&\frac {L^2}{2r^2}-\frac {GML^2}{r^3} \end {eqnarray}$$(Carroll's 7.47 & 7.48, where I've set ##\epsilon=0## for a null geodesic). Carroll says that E and L are the energy and angular momentum of the light pulse and says (his equations 7.43 & 7.44)$$\begin{eqnarray}E&=&\left (1-\frac {2GM}r\right)\frac {dt}{d\lambda}\\

L&=&r^2\frac {d\phi}{d\lambda}\end {eqnarray} $$

My question is: if I am hovering at r armed with a protractor and a laser pointer, how do I relate my protractor reading to L and E in order to determine where my beam goes?

I can rearrange (1) to get $$\frac {dr}{d\lambda}=E\sqrt {1-\frac {L^2}{E^2r^2}\left (1-\frac {2GM}r\right)} $$which tells me that E isn't important on its own - it's a scale factor, but since ##\lambda## has no particular physical significance it comes out in the wash. What is important is L/E. Dividing (4) by (3), then, I think I can write $$\frac LE =\frac r {1-2GM/r}\left (r\frac {d\phi}{dt}\right) $$The term in brackets is just the locally measured tangential component of the "muzzle velocity" of my beam. So I can replace it (in the c=1 units) with ##\cos\psi##, where ##\psi## is my protractor reading (zero points to the center of the black hole).

Is that right?

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