[saspinski (OP)]

Suppose a beam of light is sent from the earth to a (huge) artificial satellite. Between the moment the beam reaches the satellite surface, until it bounces back from a mirror at its center, and passes again by its surface, there is a time interval of 2r/c where r is the satellite radius.
For someone in the earth, the time would be bigger than 2r/c due to the satellite orbital speed. The beam should be travelling a curved path from the surface to the center and back to the surface from earth perspective.
Is that reasoning correct to represent the time dilation due to gravity? That effect would be smaller for distant orbital radius because of the smaller speed.   

[evan_au]

I assume that this question is about time dilation due to gravitational fields (ie general relativity), but also time dilation due to velocity (ie special relativity)?

Both effects affect the atomic clocks on the GPS satellites, and they work on opposite directions, the two effects partially canceling each other. The clocks on GPS satellites run faster in orbit because they are further out of Earth's gravitational well; but they also run slower because they are traveling at orbital speed.

a (huge) artificial satellite... a beam of light reaches the satellite surface (and) bounces back from a mirror at its center

From this description I am imagining something like a huge partially reflective balloon in a circular orbit around Earth, inflated by a low-pressure gas, with something like a retroreflector at its center?

A laser pulse from Earth's surface is partially reflected from the metalized surface of the balloon and the remainder is reflected back to Earth from the retroreflector at the center. Someone on Earth can measure the delay between the two light pulses.

A similar set of tools on the ground is used to measure the distance to the Moon.

Since this satellite is a big balloon, I assume that it has negligible mass, so it will effectively have no gravitational time dilation of its own (just the time dilation due to the Sun and the Galaxy, which is shared with the Earth).

Since the retroreflector is moving in a circular orbit, it is not traveling radially towards or away from the laser, so there will be no doppler shift, and no relativistic frequency shift (ie there is no laser source on the balloon).

For someone in the earth, the time would be bigger than 2r/c due to the satellite orbital speed.

I don't expect so. There is no clock on board the satellite, so we are really just talking about the speed of light in an (almost) vacuum.

I don't think the satellite orbital speed will have any impact on the result.

The beam should be travelling a curved path from the surface to the center and back to the surface from earth perspective.

I don't expect so. The photons in the laser beam travel in a straight line. Some will be reflected from the metallic surface of the balloon; others will bounce off the mirror. Because they are traveling radially out from the Earth and returning radially to the Earth, any curve will be negligible. (When Eddington was trying to test the General theory of Relativity, he used light traveling tangentially to the Sun's surface, close to the Sun's much larger mass, and was able to demonstrate a small bend in direction.)

I expect that the light will travel in straight lines, for all practical purposes.
 

For someone in the earth, the time would be bigger than 2r/c

Since the Earth is in a gravitational well, time will pass more slowly on Earth. So I expect that the measured time will be smaller than 2r/c.

But in reality, light is very fast, and gravitational time dilation on Earth is very small. The ability to measure r accurately is very limited, since a balloon is very stretchy, it's shape will be slightly distorted by the heat of the Sun, the Solar Wind and/or any drag from Earth's outer atmosphere, and it is very hard to mount a retroreflector at the precise geometric center. The proposed experiment is trying to measure something very precisely (gravitational time dilation), using something which can be known only very approximately (r).

So I suggest that the best way to measure r is to measure the time difference between a pulse of light reflected from the surface, and one measured from the mirror at the center. Forget measuring gravitational time dilation this way. Much better measurement of relativistic effects are available from GPS satellites and Gravity Probe B.

[chris]

Excellent answer. Thanks Evan

[saspinski (OP)]

What I am trying to describe is similar to that example in SR, when one observer look at a passing car where a pulse of light travels from the ground to the top, is reflected by a mirror and travels back to the ground. A man in the car measures t = 2h/c, h being the car height. The observer out of the car measures t>2h/c, because the light path is a triangle for him.

In my example, t=2r/c for an potential observer in the satellite, (diagram - right)  but for an observer in the earth the path should be greater because the satellite moves while the light is traveling inside it. (Of course the displacement is exagerated but just to show that the path would be greater).

But anyway, I had not realized that from the frame of reference of the satellite, if a pulse is sent to the earth and a device of the same radius receives it, the time calculation is reverse, because the earth is moving now. So, any effect is not related to gravity or orbital distance.

[diagram=761_0]

[evan_au]

If you want to measure relativistic effects like time dilation, and don't have a black hole handy*, you need to use some very precise measurements.

The quantity we can measure with the greatest accuracy is time, using quite compact and accurate atomic clocks (which are rapidly dropping in price).

So if you want to measure relativistic effects, putting an atomic clock on the object of interest is probably the most cost-effective way of measuring the less subtle effects predicted by relativity.

Gravitational waves are one of the more subtle effects predicted by relativity. Scientists have spent hundreds of millions of dollars on detecting gravitational waves, with no announced results yet.

*Don't experiment on black holes at home!

[saspinski (OP)]

Atomic clocks are based on the frequency of a foton, that is produced after an electron returns from an excited state, if I understood well what I read in the internet. If that frequency is a intrinsic material property, (and I donīt see why it isn't) and not affected by gravity, then they are fine to measure relativistic tiny efffects.

About a property that is affected by gravity, a pendulum clock will run slower at high altitudes and even stop in orbit, what could lead us to a different relation of time dilation and gravity, if based only on that experimental device.

About my question, I read in the web (could not post the link here) an interesting explanation: if I am in free fall, and compare the Δt of my clock with another one attached to a tall building for example, just when I pass by, it would "tic" slower because it is moving to me. Some meters below I make the same comparison to another clock. It would "tic" even slower than the first one, because my speed is now greater. That relates time dilation due to speed, to time dilation due to gravity.