Distance change due to gravity

[nurica]

Please help me with this (relatively) simple question.

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.

 

[PeterDonis]

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.

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?

I note that while laser light was traveling to the Moon and back it was traveling under influence of Earth's gravitation.

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.

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.

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.

 

[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. Let's 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.

 

[PeterDonis]

Since it is a thought experiment we can imagine that half of Earth was cut off and instantly removed.

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.

Classic test for GR was to measure the difference of star position (viewed from Earth) as it passed near the Sun. Let's 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?

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.)

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.

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?

 

[Bill_K]

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

Doppler shift as the distance changes?

 

[PeterDonis]

Doppler shift as the distance changes?

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.

 

[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.

 

[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?

 

[PeterDonis]

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.

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.)

 

[PeterDonis]

Is it that actual distance changed or just because gravity slows clocks and light takes longer to arrive?

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

For example, if we measure distance to a satellite which is on the opposite side from the Sun, using brightness method.

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

Will we see brightness reduced when satellite gets closer to the Sun?

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.

 

[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

 

[Bill_K]

Is it that actual distance changed or just because gravity slows clocks and light takes longer to arrive?

Both. See this earlier post for a derivation.

 

[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?

 

[PeterDonis]

Is there an effect similar to Shapiro delay but which displays shrinkage of distances?

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.)

 

[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".

 

[Bill_K]

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?

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.)

 

[PeterDonis]

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

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.

 

[Bill_K]

"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".

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.

 

[nurica]

It's from here: http://archive.ncsa.illinois.edu/Cyberia/NumRel/GenRelativity.html

 

[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?

 

[PeterDonis]

It's from here: http://archive.ncsa.illinois.edu/Cyberia/NumRel/GenRelativity.html

Thanks for the link. One key thing that this page doesn't tell you is that it is implicitly assuming that you have picked a particular system of coordinates, i.e., that you have picked a particular way of splitting up spacetime into "space" and "time". The embedding diagram it gives (the "bowling ball on a stretched rubber sheet" diagram), and the statement you quoted, are really only valid for one particular system of coordinates. In general, there are many different ways of splitting up spacetime into space and time, and they lead to different ways of looking at the things discussed on this web page.

 

[Agerhell]

Please help me with this (relatively) simple question.

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.

If you are on a planet with the same radius as the Earth but half the mass and you have a moon exactly like the one we have now passing by with the same velocity and distance from us as the moon have right now, things that would effect your measurements are:

The clock for you, sitting on the the planet will tick faster, due to the fact that the "gravitational well is less deep" as the planet is lighter than the earth. The light signal will travel faster due to the weaker gravitational field.

You could, in principle, attach rocket enginges to both the moon and the planet so that for the time of the experiment they both move identical to how the moon and the Earth moves now.

How fast your clock will tick is easy to calculate (at least if you ignore the influence of the moon), I do not know however, what is the correct expression for the velocity of light in coordinate time if there are two spherical symmetric massive objects closeby...

 

[nurica]

Some of the variables in the experiment can be ignored. Say we can replace the Moon with a satellite and ignore its mass.

What remains is that clocks on half-earth will tick faster, so the signal will run faster, so we'll see that the distance shrink, despite the fact the neither the Earth nor the satellite had actually changed positions, right?
On other hand, they did changed positions, but not because they moved but because the bending of space-time is changed.

I'm looking for configuration where the distance is increasing.

If the distance is increasing when we increase the mass at near location from which we make measurements, it could be used as the reason for the expansion. For example, if the amount of dark matter is constantly increasing, and we can't see it directly since it is 'dark', then all distances in the Universe must increase.

I must point here that I'm not proposing any new theories here - everything is within classic GM.

Does that sound reasonable?
I mean that if we imagine a universe with constantly increasing mass, we'd observe that it is constantly expands, without employing any dark energy. Mass increase can work as dark energy?

 

[PeterDonis]

Some of the variables in the experiment can be ignored. Say we can replace the Moon with a satellite and ignore its mass.

Sure, you can ignore its mass, but there will still be some variables you can't ignore, and you have to decide what holding those variables constant means before you can even ask the questions you're asking; and the answers to the questions will depend on how you define what holding variables constant means.

What remains is that clocks on half-earth will tick faster, so the signal will run faster, so we'll see that the distance shrink, despite the fact the neither the Earth nor the satellite had actually changed positions, right?

It depends on how you define what "changed positions" means. How do we tell whether the positions are changed or not? The measure of "distance" you are using is round-trip light travel time; but that means that the round-trip light travel time *defines* what the distance is. So you can't then ask how the round-trip light travel time would change if the distance were "held constant"; that makes no sense.

In short, the term "distance" does not have a pre-existing meaning, but you are treating it as if it does.

 

[nurica]

Let me move the latest question to a new thread so it will be more visible to newcomers