Pulsing from a ->c satellite

[Pengwuino]

Ok let's say we have a satellite going near the speed of light. The satellite is programmed to emit an e/m pulse once every second of the satellites lifetime. The speed of the satellite is an integer of a certain # of orbits/second. Let's say its at a speed of exactly 100 orbits/second (and say its a planet where 100 orbits/second is almost the speed of light). Let's also say there is a reciever picking up these signals on the ground and its programmed to aim the wave at that receiver.

How much general time (since i didnt use specific numbers) would pass on Earth before each pulse was received from an observer on earth. Also, what effect might this have on the e/m pulse? Would its frequency or wavelength change?

 

[jtbell]

The frequency with which the ground station receives the satellite's pulses, and the frequency of the radio signal, both vary with time, depending on whether the satellite is approaching or receding from the ground station, and on the angle between the satellite's velocity and the line of sight from the satellite to the ground station. You need to use a version of the relativistic Doppler effect formula that contains the angle.

 

[Pengwuino]

Well i was trying to imply that the satellite is always at teh same point relative to the ground station when the pulse is sent out

 

[pervect]

This is going to be a long calculation, be warned in advance

OK, let's make life easy and assume that the satellite is either orbiting at grouind level, or that there is a tower raised up above ground level so that we compare the satellite's clock to a clock that's not only at the same altititude, but that passes very close by the satellite on every orbit.

The "planet" is not-quite a black hole, because the photosphere where light orbits is at r = (3/2) r_s.

I don't think the orbit would be stable, but that's not a fatal objection, the satellite can have thrusters and a program designed to stabilize the unstable orbit.

The equations govering the orbit around a black hole apply directly to this case (even though the planet isn't of itself quite a black hole), because we are assuming the planet is a spherically symmetrical body, and the gravity of any such body is given by the Schwarzschild metric (the same metric as that of a black hole) above the surface of the body.

We will work out the problem in Schwarzschild coordinates, r and t. We have

r = constant

which means dr/dtau must be zero. From eq 25.16a in MTW, this means

V^2 = E^2. By 25.16b this means

eq #1
(1-2m/r) (1+L^2/r^2) = E^2

here E and L are conserved quantiites of orbits in the Schwarzschild geometry that are the relativistic equivalents of the energy per unit mass and the angular momentum per unit mass of the body. Note that the energy and angular momentum of a body in orbit are also conserved under Newtonian gravity.

Now dt/dtau_sat = E/(1-2m/r) by 25.18

here dtau_sat is the time measured on the satellites clock, and dt is an abstract quantity that has no direct relevance to the problem, the schwarzschild coordinate time interval, which can be thought of the time interval "at infinity".

Now we have to find dt/dtau_ground, the ratio of the schwarzschild coordiante time interval to the proper time of a clock on a tower. This will be

dt/dtau_grnd = 1/sqrt(1/(1-2m/r))

by the standard gravitational time dilation formula.

We can take the ratio and find

(dt/dtau_sat) / (dt/dtau_grd) = (dtau_gnd / dtau_sat) = E/sqrt(1-2*m/r)

Using the results of eq #1, we can simplify this to

dtau_gnd / dtau_sat = sqrt(1+L^2/r^2)

This means that the "ground" clock on the tower will show more time elapsed than the orbiting clock.

Also of interest is eq 25.20, which gives the local tangential velocity (ordinary velocity, not 4-velocity) of the orbiting satellite as measured by the ground-based observer. This is

v = L/(r Elocal), where Elocal = E/sqrt(1-2m/r)

I think it ought to be possible to express all the results in terms of v rather than E and L, but I haven't done the manipulations to do so. I also badly need to recheck all the steps to see if I've made any errors, the calculations are rather long.

 

[Pengwuino]

*shrugs* it was just a mental excercise in my mind lol. a lot of this stuff is seemingly beyond me unless i take some more time to look at it.

 

[pervect]

I think it's an interesting problem. I expect that the result should eventually wind up as the familiar Lorentz contraction expressed in terms of the local tangential velocity, but I haven't been able to work through all the math to see if this is in fact correct.

I can see where the details are quite technical, unfortunately if one wants to really solve a detailed problem one has to go through the math to do it. One can always try for insights to make the problem simpler. Unfortunately, ometimes one can get into trouble that way, when one makes bad assumptions that look reasonable at first glance - so sometimes its a lot safer to slog through an approach that one knows is correct than to look for a beautiful conceptual simplificaiton.

 

[Pengwuino]

I was just wondering if like, the station would receive pulses every 1 second or not and would the pulse frequency generated according to the satellite be the same frequency as the tower gets. I just realized the orbits would appear much different at such speeds so i have to take into account what an orbit even looks like when viewed from the satellite.

 

[Ich]

I was just wondering if like, the station would receive pulses every 1 second or not and would the pulse frequency generated according to the satellite be the same frequency as the tower gets. I just realized the orbits would appear much different at such speeds so i have to take into account what an orbit even looks like when viewed from the satellite.

I guess you didn´t want to consider gravity, just special relativity.
Then you will find that the station receives 1 pulse per second, and this pulse has a shifted frequency, according to the relativistic Doppler Effect.