General Relativity & The Sun: Does it Revolve Around Earth?

[PeterDonis]

I hope this is also correct

If you're going to talk about Doppler shift, you probably should also address the case where the observers are moving towards each other, which means they will see each other blueshifted--"running fast" instead of "running slow"--and time dilation only appears after this is corrected for light travel time.

Also, as my phrasing just now indicates, I'm not sure that describing the case you describe as "correcting for the redshifting effect" is a good way to describe it. It might help to look at the math. If the two observers are moving away from each other, the relativistic Doppler factor is

$$
D = \sqrt{\frac{1 - \beta}{1 + \beta}} = \frac{1}{\gamma} \frac{1}{1 + \beta}
$$

where $\gamma$ is the usual relativistic factor $1 / \sqrt{1 - \beta^2}$, so $1 / \gamma$ is the time dilation factor. Notice that the Doppler factor is a smaller number than the time dilation factor; in other words, each observer actually sees (through their telescope, say) the other's clock running more slowly than the time dilation factor alone would imply. This extra factor is because as the observers move apart, the light travel time between them increases.

If the two observers are moving towards each other, the relativistic Doppler factor is

$$
D = \sqrt{\frac{1 + \beta}{1 - \beta}} = \frac{1}{\gamma} \frac{1}{1 - \beta}
$$

Here the Doppler factor is a larger number than the time dilation factor, and in fact is greater than $1$, indicating a blueshift, not a redshift. The "correction" here, to get the time dilation factor less than $1$, so that each observer "observes" the other's clock to be running slow (when they actually see, as in through their telescope, the other's clock running fast), is due to the light travel time between the observers decreasing as they move towards each other.

My own preference is actually to take the Doppler factor as primary, since that's what is actually directly observed. The "correction" to get the time dilation factor that appears in many of the common formulas can then be explained as being due to light travel time, as I did above. Trying to explain it the other way is possible, but might be more likely to cause confusion.

 

[JohnNemo]

If you're going to talk about Doppler shift, you probably should also address the case where the observers are moving towards each other, which means they will see each other blueshifted--"running fast" instead of "running slow"--and time dilation only appears after this is corrected for light travel time.

Also, as my phrasing just now indicates, I'm not sure that describing the case you describe as "correcting for the redshifting effect" is a good way to describe it. It might help to look at the math. If the two observers are moving away from each other, the relativistic Doppler factor is

$$
D = \sqrt{\frac{1 - \beta}{1 + \beta}} = \frac{1}{\gamma} \frac{1}{1 + \beta}
$$

where $\gamma$ is the usual relativistic factor $1 / \sqrt{1 - \beta^2}$, so $1 / \gamma$ is the time dilation factor. Notice that the Doppler factor is a smaller number than the time dilation factor; in other words, each observer actually sees (through their telescope, say) the other's clock running more slowly than the time dilation factor alone would imply. This extra factor is because as the observers move apart, the light travel time between them increases.

If the two observers are moving towards each other, the relativistic Doppler factor is

$$
D = \sqrt{\frac{1 + \beta}{1 - \beta}} = \frac{1}{\gamma} \frac{1}{1 - \beta}
$$

Here the Doppler factor is a larger number than the time dilation factor, and in fact is greater than $1$, indicating a blueshift, not a redshift. The "correction" here, to get the time dilation factor less than $1$, so that each observer "observes" the other's clock to be running slow (when they actually see, as in through their telescope, the other's clock running fast), is due to the light travel time between the observers decreasing as they move towards each other.

My own preference is actually to take the Doppler factor as primary, since that's what is actually directly observed. The "correction" to get the time dilation factor that appears in many of the common formulas can then be explained as being due to light travel time, as I did above. Trying to explain it the other way is possible, but might be more likely to cause confusion.

How about this?

'Also, as measured from the Earth, time on the spaceship runs more slowly – measured from the Earth the spaceship crew are in slow motion. This is called time dilation. Again the crew of the spaceship do not feel any different – they are only in slow motion as measured from the Earth. Of course because speed is relative to the observer (we cannot say that the spaceship is in any absolute sense moving with any particular speed, or that the Earth is in any absolute sense moving with any particular speed, but only that they are moving at 400 million miles per hour relative to each other) the ground crew will be in slow motion as measured from the spaceship.

When we talk about ‘measuring’, from the Earth, how time runs more slowly on the spaceship, this does not just mean looking through a telescope with a stopwatch handy. Because the Earth and the spaceship are traveling away from each other, a clock on the spaceship will appear to be running slower anyway. Say the minute hand is showing 20 minutes past the hour, by the time the minute hand shows 21 minutes past the hour, the spaceship will be that much further away from the Earth so the light showing 21 minutes past the hour will take longer to get to the Earth than the light which showed the minute hand at 20 minutes past did, and so, for that reason alone, the clock on the spaceship will appear to run slower. This is the Doppler effect. But once you have calculated the apparent slowing down which you expect, for a spaceship moving at the speed it is, you find that it appears even slower than you expected, and identifying that extra slowing down, over and above the Doppler effect, is what we mean by ‘measuring’.

If the spaceship is traveling towards the Earth, the Doppler effect will mean that a clock on the spaceship will appear to run faster because the light showing the minute hand at 21 minutes past will take less time to get to the Earth than the light showing the hand at 20 minutes past did. But if you work out how much faster you expect it to be, because of the Doppler effect, you discover that it is not as fast as you expected because time on the spaceship has actually slowed down (relative to the Earth).'

 

[PeterDonis]

How about this?

You didn't read my previous post carefully enough.

once you have calculated the apparent slowing down which you expect, for a spaceship moving at the speed it is, you find that it appears even slower than you expected, and identifying that extra slowing down, over and above the Doppler effect, is what we mean by ‘measuring’.

This is not correct as you state it. If the spaceship is moving away from you, its clock appears, through your telescope, to be running even slower than you would calculate it to be running just due to the time dilation factor. In other words, the time dilation is not an extra slowing down over and above the Doppler effect; it is less slowing down, once you correct for the light travel time, than the Doppler effect you actually observe.

Your explanation of the other case, with the spaceship moving towards you, is too confused for me to try to correct. Please go back and read my previous post again, carefully.

 

[JohnNemo]

This is not correct as you state it. If the spaceship is moving away from you, its clock appears, through your telescope, to be running even slower than you would calculate it to be running just due to the time dilation factor. In other words, the time dilation is not an extra slowing down over and above the Doppler effect; it is less slowing down, once you correct for the light travel time, than the Doppler effect you actually observe.

How's this?

'When we talk about ‘measuring’, from the Earth, how time runs more slowly on the spaceship, this does not just mean looking through a telescope with a stopwatch handy. Because the Earth and the spaceship are traveling away from each other, a clock on the spaceship will appear to be running slower anyway. This is called the Doppler effect. Part of the Doppler effect is caused by differences in how long light takes to reach the Earth. Say the minute hand is showing 20 minutes past the hour, by the time the minute hand shows 21 minutes past the hour, the spaceship will be that much further away from the Earth so the light showing 21 minutes past the hour will take longer to get to the Earth than the light which showed the minute hand at 20 minutes past did, and so, for that reason, the clock on the spaceship will appear to run slower. But once you have calculated the apparent slowing down which you expect, for a spaceship moving at the speed it is, with differences in how long light takes to get to the Earth, you find that that the clock on the spaceship appears even slower than you expected, and identifying that extra slowing down, over and above the slowing down expected because of varying light transit time, is what we mean by ‘measuring’.

 

[PeterDonis]

How's this?

Still wrong. You have not grasped that the amount of "slowing down" you get due to the time dilation factor is less than the "slowing down" you actually see through your telescope, which is what "Doppler shift" means.

You also continue to ignore the case where the spaceship moves towards you, and the Doppler effect you see through your telescope is a blueshift--speeding up, not slowing down.

 

[JohnNemo]

Still wrong. You have not grasped that the amount of "slowing down" you get due to the time dilation factor is less than the "slowing down" you actually see through your telescope, which is what "Doppler shift" means.

So, as I understand it, the whole phenomenon, as observed through the telescope, is called the Doppler shift. The reason for the Doppler shift is two contributing effects

1. varying light transit time

2. time dilation

You also continue to ignore the case where the spaceship moves towards you, and the Doppler effect you see through your telescope is a blueshift--speeding up, not slowing down.

True, but only a temporary ignoring while I try to get my head round the redshift.

 

[PeterDonis]

as I understand it, the whole phenomenon, as observed through the telescope, is called the Doppler shift.

Yes.

The reason for the Doppler shift is two contributing effects

1. varying light transit time

2. time dilation

That's one way of looking at it, yes. The other way of looking at it is that Doppler shift is the fundamental thing (since that's what is actually observed), and "time dilation" is due to two contributing effects: Doppler shift and varying light travel time.

The first viewpoint (the one you described in the above quote) is going to seem more natural if you're used to thinking about SR in terms of inertial frames. The second (the one I described in the previous paragraph) is likely to seem more natural if you're used to thinking about SR in terms of spacetime geometry. I favor the second approach, but both are mathematically equivalent, so it's really a matter of personal preference.

 

[JohnNemo]

Yes.
That's one way of looking at it, yes. The other way of looking at it is that Doppler shift is the fundamental thing (since that's what is actually observed), and "time dilation" is due to two contributing effects: Doppler shift and varying light travel time.

The first viewpoint (the one you described in the above quote) is going to seem more natural if you're used to thinking about SR in terms of inertial frames. The second (the one I described in the previous paragraph) is likely to seem more natural if you're used to thinking about SR in terms of spacetime geometry. I favor the second approach, but both are mathematically equivalent, so it's really a matter of personal preference.

The reason why I wanted to mention the Doppler effect in my write-up when talking about Special Relativity is that when someone is learning SR it is tempting to think that the phenomena are just illusions and that behind it all there is something non-relative which is 'actually' happening. So I wanted to make clear that time dilation is a real thing and should not be confused with varying light transit time which is only responsible for an appearance of slowness - an optical illusion. I was inclined to call this varying light transit time effect the Doppler effect. Is it wrong to do that or is the word capable of being used in that restricted sense? If it is wrong, is there some other word which describes the varying light transit effect (isolated from the time dilation effect).

 

[PeterDonis]

when someone is learning SR it is tempting to think that the phenomena are just illusions and that behind it all there is something non-relative which is 'actually' happening

And that is correct. The thing that is non-relative that is actually happening is spacetime and events in spacetime, such as you receiving light from some source with a particular redshift/blueshift at some event along your worldline. "Time dilation" is what might be termed an "illusion", since it's an artifact of your choice of coordinates (see below).

Relativity does not say "everything is relative". In fact Einstein at one point said the theory was misnamed, and should have been called the "theory of invariants"--the things that do not change when you change your choice of coordinates.

I wanted to make clear that time dilation is a real thing and should not be confused with varying light transit time which is only responsible for an appearance of slowness - an optical illusion

But here's the thing: time dilation is only a "real thing" if you say that things that depend on your choice of coordinates are "real things"; and doing that is problematic in relativity, since relativity says your choice of coordinates doesn't affect any physics. Whereas the actual observed redshift/blueshift in light coming from some other observer is a "real thing" in the simplest sense: it's what you directly observe.

 

[JohnNemo]

But here's the thing: time dilation is only a "real thing" if you say that things that depend on your choice of coordinates are "real things"; and doing that is problematic in relativity, since relativity says your choice of coordinates doesn't affect any physics. Whereas the actual observed redshift/blueshift in light coming from some other observer is a "real thing" in the simplest sense: it's what you directly observe.

If we only do one experiment then I can see an argument could be made that the single observation of redshift/blueshift is all there is. But if we do many experiments we can show that redshift/blueshift is related to the speed at which the object is approaching/receding, in our direction (single dimension), and that there is a separate element - time dilation - which is related to the object's speed in 3D. So having done all the experiments with objects traveling at different angles and produced our mathematical theory which we believe models reality, is it not the case that we can then distinguish the time dilation element from the varying light transit time element?

 

[PeterDonis]

if we do many experiments we can show that redshift/blueshift is related to the speed at which the object is approaching/receding

How do you measure the speed? You are aware that the standard way of doing this for distant objects is...using the Doppler shift?

In other words, this "speed" you talk about is not an independent variable in the usual case; it's just another way of describing the observed redshift/blueshift. To make it an independent variable, you would need to make other independent measurements of it, and that would require having a fleet of observers distant from you whose clocks were synchronized with yours and who were verified to be at rest relative to you, who could make local measurements of the object that is emitting the light you are receiving. In other words, to set up an actual concrete "inertial frame" centered on you.

If you do this, then yes, you can separate out "light travel time" from "time dilation", but now "time dilation" doesn't mean the same thing as it meant before. It now means, not a coordinate-dependent quantity that you calculate, but actual measurements that your fleet of observers is making, comparing their clock readings with those of the object emitting the light. Similarly, "light travel time" is now not something you calculate, but something your fleet of observers directly measures (by recording their clock readings as the light rays from the object pass them).

 

[JohnNemo]

@JohnNemo the corrections you made look fine, but I did spot one other item:

"if a mass is accelerating that adds to the distortion"

What are you referring to here?

I was basing this on https://en.wikipedia.org/wiki/Frame-dragging but I am not sure about this. Is it just acceleration or any movement which adds to the distortion?

 

[PeterDonis]

I was basing this on https://en.wikipedia.org/wiki/Frame-dragging but I am not sure about this. Is it just acceleration or any movement which adds to the distortion?

Neither. Frame dragging, which can be thought of as an extra "distortion" to spacetime, yes, is due to the source of gravity having nonzero angular momentum, where "angular momentum" here is has a technical definition that is probably closest to "rotating with respect to the distant stars" of the concepts we've discussed--and where "rotating" here is to be taken in the intuitively obvious sense that someone "at rest" on the rotating object will see a given distant star as moving in a big circle around his sky instead of staying at a single fixed point in his sky.

This concept is different from any of the indicators of rotation we have discussed in this thread. For example, a black hole can be rotating in this sense, and a black hole is a vacuum solution, so there is no matter anywhere and therefore nothing that can have any proper acceleration or "motion" of any kind. It's all just spacetime geometry.

 

[JohnNemo]

Neither. Frame dragging, which can be thought of as an extra "distortion" to spacetime, yes, is due to the source of gravity having nonzero angular momentum, where "angular momentum" here is has a technical definition that is probably closest to "rotating with respect to the distant stars" of the concepts we've discussed--and where "rotating" here is to be taken in the intuitively obvious sense that someone "at rest" on the rotating object will see a given distant star as moving in a big circle around his sky instead of staying at a single fixed point in his sky.

This concept is different from any of the indicators of rotation we have discussed in this thread. For example, a black hole can be rotating in this sense, and a black hole is a vacuum solution, so there is no matter anywhere and therefore nothing that can have any proper acceleration or "motion" of any kind. It's all just spacetime geometry.

https://en.wikipedia.org/wiki/Frame-dragging#Effects also mentions linear frame dragging. Is this also recognised n GR?

 

[PeterDonis]

linear frame dragging

I haven't seen anything about this in the GR textbooks and papers I've read. Unfortunately the Wikipedia article does not give a link to the Einstein lecture where he apparently mentioned it.

Offhand I would not expect an "similar" effect for linear momentum as opposed to angular momentum, for the same (heuristic) reasons as linear motion relative to the distant stars is not locally detectable the way that angular motion (rotation) relative to the distant stars is (putting aside all the subtleties we've been discussing about that). But it is true that for an object moving at high speed past a gravitating mass, GR makes different predictions than Newtonian gravity; it could be that this is what "linear frame dragging" is supposed to refer to.

 

[JohnNemo]

Neither. Frame dragging, which can be thought of as an extra "distortion" to spacetime, yes, is due to the source of gravity having nonzero angular momentum, where "angular momentum" here is has a technical definition that is probably closest to "rotating with respect to the distant stars" of the concepts we've discussed--and where "rotating" here is to be taken in the intuitively obvious sense that someone "at rest" on the rotating object will see a given distant star as moving in a big circle around his sky instead of staying at a single fixed point in his sky.

This is where I am having difficulty understanding because you often say (e.g. In #152) that a region of space a long way from the nearest star is flat (because the matter and energy in the universe is spherically symmetrical) but here we appear to have a distortion of spacetime caused by the distant stars moving in a big circle.

 

[Nugatory]

This is where I am having difficulty understanding because you often say (e.g. In #152) that a region of space a long way from the nearest star is flat (because the matter and energy in the universe is spherically symmetrical) but here we appear to have a distortion of spacetime caused by the distant stars moving in a big circle.

Frame dragging only appears in the egregiously non-flat regions of spacetime near a massive rotating body; the distant stars are completely irrelevant to the spacetime curvature in such a region.

The only reason we mention the distant stars is that we said "near a massive rotating body" so we have be clear about what we mean by "rotating", and in this context an intuitive definition of "rotating" involves the distant stars.

 

[JohnNemo]

Frame dragging only appears in the egregiously non-flat regions of spacetime near a massive rotating body; the distant stars are completely irrelevant to the spacetime curvature in such a region.

The only reason we mention the distant stars is that we said "near a massive rotating body" so we have be clear about what we mean by "rotating", and in this context an intuitive definition of "rotating" involves the distant stars.

Is the bulge around the Earth’s equator caused by this type of frame dragging effect?

 

[Nugatory]

Is the bulge around the Earth’s equator caused by this type of frame dragging effect?

No. That's a classical phenomenon, well understood since Newton.
Googling for "equatorial bulge cause" will find some good explanations.