# Distance change due to gravity

1. Apr 22, 2013

### nurica

Experiment 1: I use laser method to measure distance from my particular location on the surface of Earth to the Moon. The distance I got is D. I note that while laser light was traveling to the Moon and back it was traveling under influence of Earth's gravitation.

Experiment 2: With all other parameters unchanged (like bodies locations etc), Earth suddenly lost half of its mass. The loss of mass was sudden and the Moon had no time to adjust its orbit, so everything else is the same except Earth's mass. Now I make measurement again. The distance I got is D1.

Earth's mass lower - gravitation lower - clocks tick faster. Will I see D1 smaller than D or the opposite? Which formula can I use to calculate the difference?

Thank you.

2. Apr 22, 2013

### Staff: Mentor

In other words, you measure the round-trip travel time $T$, and then use the formula:

$$D = \frac{c T}{2}$$

where $c$ is the speed of light. Correct?

Yes, this is true, but it doesn't affect the above formula; if you're measuring distance "using the laser method" (this distance is usually called "radar distance" in relativity texts), you are *defining* the distance by the above formula. Whether or not gravity is present is a different, and unrelated, question.

This is not possible. Where did the mass go? Mass can't just disappear; that violates local energy-momentum conservation.

You could consider a scenario where we have an object identical to the Earth's Moon, orbiting a planet with half the mass of the Earth, such that at some instant "everything else is the same" as in Experiment 1. But then you would have to define what "everything else the same" *means*. Does it mean the center-to-center distance between the planet and the moon is the same? How do we measure this distance?

Basically, there is no way to just change the mass of the planet in this scenario and leave everything else "the same"; more precisely, there is no way to *define* what "everything else the same" means in such a scenario. So your question doesn't have a well-defined answer. If you want to try to get a handle on how gravity affects "distance", you're going to have to come up with a different scenario.

3. Apr 22, 2013

### nurica

Since it is a thought experiment we can imagine that half of earth was cut off and instantly removed. Important is the gravitational force that changed and so the space distortion is changed and how it affects the calculations.

Ok, here is more realistic experiment: Classic test for GR was to measure the difference of star position (viewed from Earth) as it passed near the Sun. Lets keep all the settings of that experiment but measure something else - a distance to the star (star A). If we ignore the change of the Earth's orbit and make two measurements: one is when the Sun is far away from path between Earth and star A (distance D); another is when star A is visible very close to the Sun's disk (distance D1).

Will D be different from D1 and in which way?

As to the method we use to measure the distance, we can assume that star A is a pulsar, and we measuring it's frequency. We record the frequency from time when the pulsar is away from the Sun and wait until the pulsar passes near the Sun (as we see it from Earth; actual pulsar is in another galaxy). If frequency changes then proximity to the Sun did affect the measured distance.

Thank you.

4. Apr 22, 2013

### Staff: Mentor

No, we can't, because it violates the laws of physics. You can't imagine a thought experiment that violates the laws of physics. Or, rather, you can, but then there's no point in asking what the laws of physics say about what happens.

Yes, this is a better scenario. The answer is that D1 will be larger than D, if we use "radar distance" as the method of measurement, i.e., measure the round-trip travel time and use the formula I gave in my last post. (We probably want to use something closer than a pulsar for this method; Solar System measurements have been made using signals bounced off other planets. See here.)

Huh? How do you figure that?

First of all, it doesn't work: GR predicts that the pulsar frequency will be the same for both measurements.

Second, what does the frequency have to do with distance? I understand how one would measure distance by using round-trip light travel time; but how would one measure distance using frequency of the signal?

5. Apr 22, 2013

### Bill_K

Doppler shift as the distance changes?

6. Apr 22, 2013

### Staff: Mentor

But it doesn't; at least, not as I understand the scenario. We are not talking about measuring relative velocity; we are talking about measuring distance in a static situation, where we and the object in question are at rest relative to each other.

7. Apr 22, 2013

### Bill_K

As the path of the light ray nears the sun, I'd say the Shapiro delay will postpone the "ticks" coming from the pulsar. Although this is usually expressed as a time delay, it's a symptom of the extra distance the light ray has to travel.

8. Apr 22, 2013

### nurica

Great.
Is it that actual distance changed or just because gravity slows clocks and light takes longer to arrive?
For example, if we measure distance to a satellite which is on the opposite side from the Sun, using brightness method. Will we see brightness reduced when satellite gets closer to the Sun?

9. Apr 22, 2013

### Staff: Mentor

Ah, I see; as the pulsar comes near the Sun, we would see a brief period of Doppler redshift as the distance lengthens slightly; then, as it moves away from the Sun, we would see a brief period of Doppler blueshift as the distance shortens again.

I can see how this would indicate a *change* in distance; but how does it indicate the distance itself? (Even translating the Doppler shift measurements into a measurement of the total *change* in distance seems to me to require some care.)

10. Apr 22, 2013

### Staff: Mentor

Yes. These aren't different possibilities for how reality is; they are different ways of describing the same reality.

This is yet *another* method of measuring distance, which can give different results from the radar (round-trip light travel time) method. See below.

I'm not sure. Roughly speaking, the round-trip light travel time is a measure of "distance" along a curve in space from the source to the observer, whereas the apparent brightness is a measure of the surface area of a 2-sphere centered on the source that intersects the observer. The relationship between these two things gets complicated in the presence of gravity, because you can no longer assume that space is Euclidean.

11. Apr 23, 2013

### nurica

Here is a quote: "the equations of relativity predict that gravity, or the curvature of Space-Time by matter, not only stretches or shrinks distances (depending on their direction with respect to the gravitational field) but also will appear to slow down or “dilate” the flow of time".

In the experiment above (measuring distance by radar from earth to a satellite), in which direction the distance might shrink (assuming the quote is correct)? Is it when we have the Sun behind us and the satellite in front?

Thank you

Last edited: Apr 23, 2013
12. Apr 23, 2013

### Bill_K

Both. See this earlier post for a derivation.

13. Apr 23, 2013

### nurica

Is there an effect similar to Shapiro delay but which displays shrinkage of distances?
Or the formula is the same (http://en.wikipedia.org/wiki/Shapiro_delay) but replacing the unit vector in it with orthogonal one will make dt negative i.e. space will shrink in that direction?

14. Apr 23, 2013

### Staff: Mentor

How do you measure "shrinkage of distances" other than by changes in the round-trip light travel time? You originally specified the "laser method" for measuring distances, which means that the Shapiro time delay *is* a measure of change in distance. (But it's a "stretching" distance, not a "shrinking" one, since it's a time delay, not a time shortening.)

15. Apr 23, 2013

### nurica

That's correct but in the quote below it says that in some directions distances shrink. Wondering - in which.

"the equations of relativity predict that gravity, or the curvature of Space-Time by matter, not only stretches or shrinks distances (depending on their direction with respect to the gravitational field) but also will appear to slow down or “dilate” the flow of time".

16. Apr 23, 2013

### Bill_K

It will get brighter, a fact which is used all the time in gravitational lensing observations of distant galaxies. See here. (Apparently this excerpt is taken from Bernie Schutz's book.)

17. Apr 23, 2013

### Staff: Mentor

Quote from where? I can't find this on the Shapiro time delay Wiki page. If you're quoting from somewhere, it helps greatly to give a link so people can see the source and the context of the quote.

18. Apr 23, 2013

### Bill_K

This is a very general statement. A gravitational wave, for example, will alternately "shrink and stretch". But I don't see how it could be applied to the Schwarzschild solution.

19. Apr 23, 2013

### nurica

20. Apr 23, 2013

### nurica

The statement is general but how can we prove or disprove it? Without gravitational waves. My initial thought is that since gravitation delays time, it will delay it in any direction. Only in the totally flat space (if we can find such place) the light will propagate with zero delay. Any mass, in any configuration and direction, will delay light propagation. Of course the speed of light is the same, but the distances from any point to any other point can only increase in the presence of matter. So the quote must be incorrect?