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

[JohnNemo]

No. But the distribution of matter and energy can make things look simpler in a particular reference frame. You are confusing yourself by focusing on frames instead of on physics.

Consider a spherical region of space centered on the solar system, with a radius large enough to contain all the planets. As I said in a previous post, the average distribution of matter and energy in the universe is, to a good approximation, spherically symmetric about this region; that means that, if there were no matter and energy at all inside the region, spacetime in the region would be flat.

But there is matter and energy inside the region: the Sun and the planets. (There is other stuff too--satellites, asteroids, comets, etc.--but we can ignore it here.) So spacetime in the region is not actually flat; but because of the theorem I mentioned in a previous post, when figuring out the spacetime geometry within the region, we only need to consider the matter and energy inside the region.

And more than 99 percent of that matter and energy is contained in the Sun; that means that the spacetime geometry within the region of the solar system is, to a good approximation, the geometry of a single source of gravity, the Sun, in which all the other objects move on geodesics. That being the case, the simplest frame in which to describe motion in the region of the solar system is a frame in which the Sun is at rest.

But there are multiple possible frames in which the Sun is at rest--frames with different rates of rotation relative to the distant stars. Which one makes motion in the region of the solar system look simplest? You can probably guess the answer: the frame that is not rotating relative to the distant stars. (One way to see why this is the case is to imagine that the solar system was not there and the spherical region we have been talking about was empty; then spacetime in that region would be flat, and a frame not rotating relative to the distant stars would correspond to a standard inertial frame in Minkowski spacetime, which is the simplest frame in which to describe geodesic motion in flat spacetime.)

None of this means that a frame in which the Sun is at rest, and which is not rotating relative to the distant stars, is "special" in the sense of being picked out by the laws of physics. The laws of physics look the same in any frame. But the practical description of motion in the region of the solar system--what you get when you work out the particular solution of the laws of physics that describes the matter and energy and spacetime geometry in this region--looks simplest in the frame in which the Sun is at rest and which is not rotating relative to the distant stars, because of the particular distribution of matter and energy and the particular spacetime geometry it gives rise to.

I am trying to understand this. I can understand that the average distribution of matter and energy in the universe is spherically symmetric and that this would make spacetime in the region 'flat' (whatever that means).

But why (and how) is it different if you use a reference frame which is rotating relative to the distant stars?

 

[PeterDonis]

this would make spacetime in the region 'flat' (whatever that means).

It means flat except for the effects of the Sun and other bodies within the region; so if there were no bodies at all in the region, the spacetime geometry would be just like that of SR.

why (and how) is it different if you use a reference frame which is rotating relative to the distant stars?

It doesn't change the spacetime geometry, but it changes how simple the motions of objects in the solar system look in the particular coordinates you have chosen. In other words, it doesn't change the physics (your choice of coordinates can't change the physics), but it does change how easy it is to calculate what the physics actually says.

 

[JohnNemo]

It means flat except for the effects of the Sun and other bodies within the region; so if there were no bodies at all in the region, the spacetime geometry would be just like that of SR.
It doesn't change the spacetime geometry, but it changes how simple the motions of objects in the solar system look in the particular coordinates you have chosen. In other words, it doesn't change the physics (your choice of coordinates can't change the physics), but it does change how easy it is to calculate what the physics actually says.

I understand that as you put it, but in #146 @Nugatory tells me that ‘common centre of mass’ and the ‘point about which the orbits occur’ will not be the same point in many frames (e.g. in a frame in which the Earth is at rest) but they will correspond in a frame in which the fixed stars are at rest, so that appears to me to be a significant result - something about physics rather than just about ease of calculation.

 

[PeterDonis]

in #146 @Nugatory tells me that ‘common centre of mass’ and the ‘point about which the orbits occur’ will not be the same point in many frames (e.g. in a frame in which the Earth is at rest) but they will correspond in a frame in which the fixed stars are at rest, so that appears to me to be a significant result - something about physics

Ah, I see. I think I would rephrase what @Nugatory said as follows to make it clear what is actual physics:

For an isolated system in an otherwise empty region surrounded by a spherically symmetric distribution of matter, the system as a whole can be described by a "center of mass" worldline which is a geodesic. Since it is a geodesic, and since it is in a region of spacetime which, excluding the effect of the isolated system itself, is flat, that worldline defines an inertial frame in the sense of special relativity, at least throughout the empty (except for the isolated system) region: the worldline itself defines the "time axis" of the inertial frame, and any set of three mutually orthogonal, non-rotating (in the sense of zero vorticity) spacelike vectors that are all orthogonal to the worldline can be used to define the spatial axes. This frame will, by construction, be non-rotating relative to the fixed stars; and in this frame, the "common center of mass" and the "point about which the orbits occur" will, by construction, be the same (since they will both be points on the geodesic worldline that defines the time axis).

 

[JohnNemo]

That particular frame is inertial

When you say 'inertial' do you mean inertial as understood in SR? i.e. that if, absent the Sun and the Earth, you placed two particles (not subject to any forces) with negligible mass in this region of space, they would either remain at rest relative to each other or would have a constant velocity.

 

[JohnNemo]

Thank you to everyone who has posted and tried to help my understanding. I have learned at lot (I think). Having got at least a few things clear in my mind, and aware that I might forget them in a month or two’s time, I though I would write out what I think I know, and I have found this easiest to do in a sort of brief guide written as if to someone else, but its main use will be for me to read it in the future to remember what I discovered. So here it is and if I have got anything wrong, please tell me.



What is relative in General Relativity?The theory of Special Relativity is fascinating. To get a basic understanding of it you only need a good book, basic algebra, and a willingness to have your intuitive ideas about time, space and motion upset.

Special relativity is built on three ideas

· Speed is relative

· Except the speed of light which is everywhere the same

· Nothing can travel faster than the speed of light

If you are walking at about 3 mph inside a railway carriage, walking towards the front, but the train in traveling at 100 mph, we might be inclined to say that your ‘real’ or ‘absolute’ speed is about 103 mph, but actually all we can say is that your speed is about 103 mph relative to the Earth. If we chose to measure your speed relative to the Sun it would be different again, or if it is measured relative to a distant galaxy it will be different again. It turns out that there is no such thing as ‘real’ or ‘absolute’ speed: you can only measure the speed of an object relative to some other object.

The speed of light is 671 million miles per hour. Suppose a spaceship is traveling at 400 million miles per hour away from the Earth. Some time ago the spacecraft launched a smaller craft which has picked up speed and is now traveling at 350 miles million miles per hour relative to (and in front of) the mothership. We would expect that the smaller craft would be traveling at 750 million miles per hour relative to the Earth, but it turns out that that is not the case: It cannot be the case because nothing can travel faster than the speed of light.

If B is traveling at speed S relative to A, and C is traveling at speed T relative to B (in the same direction) then we expect the speed of C relative to A to be S + T. But it turns out that this is not correct and C’s speed relative to A is actually

$$\frac {S +T} {1 + ST/c^2}$$

where c is the speed of light.

If we do the mathematics it turns out that the smaller craft is traveling at 553 million miles per hour relative to the Earth, still less than the speed of light.

This is a strange result and there are other strange results of the theory of Special Relativity. It turns out that, measured relative to the Earth, the spaceship is shorter than it was when it was at rest on the Earth. This phenomenon is known as the Lorentz contraction. The crew of the spaceship do not notice any difference – from their frame of reference the length of the spaceship is the same as it has always been, but, when measured from the Earth, it is shorter.

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.General Relativity

Special Relativity is called ‘special’ because it applies to the ‘special case’ of objects which are moving at a constant velocity relative to one another (or at rest relative to each other) – i.e. the formulas it provides do not work where a body is accelerating or decelerating. If you have studied Special Relativity and are now about to look at General Relativity, it is natural to assume from the name that General Relativity must be based on the idea that everything, including acceleration, is relative. But this is not the case and the name General Relativity is potentially misleading. It is important, when considering General Relativity, to get clear in you mind what is, and is not, relative.

But, first of all, let us talk about space-time. Before Einstein, there was some debate about what space is – is it a real thing or is it just the absence of anything. In General Relativity space, or rather space-time (four dimensions including time) is a real physical thing. You can imagine it as a grid – a grid which is distorted where there is mass and/or energy. It is distorted most next to large masses and if a mass is accelerating that adds to the distortion. Now, in General Relativity, gravity is not a ‘force’ but is explained by the distortion of spacetime. There are natural paths in spacetime which any object which is not being pushed or pulled by any force will follow – these are called geodesics. The presence of mass distorts spacetime so that geodesics tend to curve towards the mass. Thus the reason why objects tend to move towards mass is not because of some force emanating from the mass but because spacetime has been curved by the mass.

So if gravity is not a force, why does it feel to us like a force? Imagine that someone lifts up a coin and drops it. When the coin is in mid air, moving towards the ground, it is moving on a geodesic taking it towards the centre of the Earth, but when the coin hits the ground, the force from the ground prevents it moving any further towards the centre of the Earth. The force which we think of as gravity is actually not a force pulling us down but a force pushing us up!

Here is another illustration. Suppose you are in a spaceship somewhere in deep space a long way from the nearest star, just drifting because the engines are switched off. You switch on the engines and the spaceship starts accelerating at, say, 9.8 metres per second per second. You feel yourself being pulled back against the cabin wall/floor towards the rear of the spaceship, but although it feels like that, you are not actually being pulled back at all: you are being pushed forwards by force of the cabin wall/floor which is (together with the rest of the spaceship) accelerating forwards.

It is the same when you are standing on the Earth, the force of the Earth is pushing you in an upwards direction and causing you to accelerate at 9.8 metres per second per second. This acceleration, measured against spacetime (which is a real physical thing, remember) is called proper acceleration. You can measure proper acceleration using an instrument called an accelerometer.

You almost certainly already possesses an accelerometer because there will be one inside your mobile phone. Your phone uses it to, for example, turn the display to landscape as you rotate the phone. You can download an accelerometer app which will actually display the proper acceleration. If you hold the phone still it will show an acceleration of 9.8 metres per second per second. Notice that when you start to move the phone the proper acceleration rate shoots up but then goes down again. This is because it is measuring acceleration – the rate of change of velocity – it is not measuring velocity itself. Any constant movement of the phone – whether fast constant movement or slow constant movement – is not shown on the accelerometer: It is only changes in velocity which show up.

In General Relativity you can choose any reference frame (including a rotating frame) and measure a body’s acceleration from that frame. This measurement of acceleration is called co-ordinate acceleration. The co-ordinate acceleration of a body can be different when measured from different reference frames. However it is important to realize that, irrespective of the reference frame, the proper acceleration of a body is invariant. Whatever reference frame you are in you can get out your binoculars and look at the reading on an accelerometer on that body and it will be whatever it is.

It is important to get this clear because if you have studied Special Relativity you will have made the mental leap from thinking about velocity as being absolute to realising that velocity is entirely a matter of velocity relative to a reference frame, and you might assume that in General Relativity acceleration is entirely a matter relative to a reference frame, but this is not the case. Proper acceleration is invariant because it is measured against the local spacetime geometry. There is no equivalent for velocity because the geometry of spacetime does not allow velocity itself to be measured against it.

If you have not come across the phrase proper velocity, you can skip this paragraph. If you have come across the idea of proper velocity you may be thinking that this is the equivalent – for velocity – of proper acceleration, but this is not really so: Proper velocity relative to an observer divides observer-measured distance by the time elapsed on the clocks of the traveling object, so it is still a relative measurement and is not (unlike proper acceleration) invariant.Rotation

Rotation does not have a well defined meaning in General Relativity. Part of the reason for this appears to be that if rotation is traditionally thought of as acceleration towards an axis coupled with velocity perpendicular to the direction of acceleration, it consists of a mixture of invariant and relative elements. The best I have been able to understand how General Relativity treats rotation is that there are four invariant indicators (which, strangely, are not necessarily all present together) which roughly equate to ‘rotation’. They are (1) pattern of proper acceleration; (2) precession; (3) Sagnac effect; (4) vorticity.Ptolemy and Copernicus

In the Middle Ages it was thought, following Ptolemy, that the Earth was fixed immovable at the centre of the universe, and the Sun orbited it. Then, at the end of the Middle Ages Copernicus, proposed that, on the contrary, the Sun was fixed immoveable at the centre of the universe and the Earth orbited the Sun. Now that we are starting to learn about Relativity it is a fascinating exercise to reassess this controversy and see who (if anyone) was right. If this does not interest you, you can stop reading now – you can learn General Relativity perfectly well without knowing anything about this historical controversy, but, if you are interested, looking at this controversy will help to apply and consolidate some of the basic features of General Relativity as discussed above.

First a reminder about what the controversy was about. Copernicus showed that you can model the motions of planets in relation to the Sun and that this is much simpler than modelling them in relation to the Earth. This insight was generally welcomed as useful irrespective of whether the Earth actually moved. For example, Tycho Brahe (1546 to 1601) combined belief in the immovability of the Earth with use of Copernicus’ calculations. When Copernicus’ book De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) was published, in 1543, the book started with an unattributed letter actually written by the Lutheran preacher Andreas Osiander who had been responsible for supervising the printing and publication. This letter, whose inclusion in the book was probably not authorised by Copernicus, was clearly designed to emphasise the uncontroversial mathematics, and deflect criticism of the controversial matter of whether the Sun or the Earth moves (Copernicus’ views on the latter had already received criticism from the Lutheran leaders, Philip Melanchthon and Martin Luther himself).

“There have already been widespread reports about the novel hypotheses of this work, which declares that the Earth moves whereas the sun is at rest in the centre of the universe Hence certain scholars, I have no doubt, are deeply offended and believe that the liberal arts, which were established long ago on a sound basis, should not be thrown into confusion. But if these men are willing to examine the matter closely, they will find that the author of this work has done nothing blameworthy. For it is the duty of an astronomer to compose the history of the celestial motions through careful and expert study. Then he must conceive and devise the causes of these motions or hypotheses about them. Since he cannot in any way attain to the true causes, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry for the future as well as for the past. The present author has performed both these duties excellently. For these hypotheses need not be true nor even probable. On the contrary, if they provide a calculus consistent with the observations, that alone is enough...”And Copernicus himself, in De revolutionibus, addresses the controversy head on. He starts with the arguments against the Earth moving relied on by ancient philosophers such as Aristotle and Ptolemy. Everything is drawn towards the centre of the Earth and would be at rest at the centre – if not checked by the surface of the Earth. Therefore the entire Earth is at rest. The Earth is heavy and not apt to move. If such a heavy object did move its motion would be violent. Copernicus deals with these arguments and then adds some arguments in favour of the Earth’s movement – e.g. the Earth is a sphere and it is natural for spheres to move in a circle.For Copernicus the controversy is over whether the Earth moves or the Sun moves – the possibility that both move being discounted:

“Hence I feel no shame in asserting that this whole region engirdled by the moon, and the centre of the earth, traverse this grand circle amid the rest of the planets in an annual revolution around the sun. Near the sun is the centre of the universe. Moreover, since the sun remains stationary, whatever appears as a motion of the sun is really due rather to the motion of the earth...

All these statements are difficult and almost inconceivable, being of course opposed to the beliefs of many people. Yet, as we proceed, with God’s help I shall make them clearer than sunlight, at any rate to those who are not unacquainted with the science of astronomy...”

So what does General Relativity tell us about who was right? Does the Earth move or does the Sun move?

The first thing to say is that the fact that the mathematics of General Relativity allows you to choose any frame of reference, including a rotating frame, including the frame of the Sun or the frame of the Earth, and describe phenomena as measured from that reference frame, is a bit of a red herring. That just shows that the mathematics is very versatile, but it does not signify anything about the physics of General Relativity (just as Copernicus’ calculations did not of themselves prove the matter one way or another).

On a large scale the matter and energy in the universe is isotropic so that if you have a region of space some distance away from the nearest star, such as our solar system, the matter and energy about that region is spherically symmetric, and the spacetime in that region would be ‘flat’ if it were empty. Consequently the spacetime geometry of the region is entirely defined by its contents - our Sun and the planets, and because over 99 per cent of the matter and energy is contained in the Sun, the spacetime geometry will, to a good approximation, be the geometry of a single source of gravity, the Sun, with all the planets moving on geodesics. A consequence of this is that the movement of the planets in a reference frame which is non-rotating relative to the distant stars and in which the Sun is at rest, is a much more regular movement (almost circular) than the movement of the planets in any reference frame in which the Earth is at rest. But this just means that the mathematics is simper: it does not help us decide which one is actually moving.

What about the invariant indicators of rotation referred to earlier? After all a body which is orbiting should exhibit at least some of these. Can we use these to determine whether the Sun orbits the Earth or the Earth orbits the Sun? Apparently not because these invariant indicators of rotation are present in both the Sun and the Earth.

So we have to conclude that Copernicus and Ptolemy were both wrong (or both half right depending how you look at it).

The final word goes to Einstein:

"Can we formulate physical laws so that they are valid for all CS (=coordinate systems), not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any CS. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS could be used with equal justification. The two sentences, 'the sun is at rest and the Earth moves', or 'the sun moves and the Earth is at rest', would simply mean two different conventions concerning two different CS. Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!"

 

[PeterDonis]

I am not sure how to post fomulas on the forum

See here:

https://www.physicsforums.com/help/latexhelp/

 

[PeterDonis]

@JohnNemo overall this looks like a good summary. There are just a couple of points that need correction:

Special Relativity is called ‘special’ because it applies to the ‘special case’ of objects which are moving at a constant velocity relative to one another (or at rest relative to each other) – i.e. the formulas it provides do not work where a body is accelerating or decelerating.

This is not correct. Special relativity is "special" because it only works if there is no gravity, i.e., if spacetime is flat. But in flat spacetime, SR can handle accelerating objects and accelerating frames just fine.

This is because it is measuring acceleration – the rate of change of velocity

Proper acceleration is not the rate of change of velocity. It is best thought of as applied force (and as you say, gravity is not a force in GR so it doesn't count here) divided by the object's mass. So when you move your phone around and the accelerometer number changes, it's not telling you how the phone's speed changes; it's telling you how much force you are applying to move the phone (and of course this is in addition to the force of the Earth pushing you up and transmitted through you to the phone).

It just so happens that, in a frame of reference fixed to the Earth, you can, with an appropriate choice of units, make the accelerometer number equal to the rate of change of velocity. But that correspondence is frame-dependent; in a different frame it won't be there. But the proper acceleration and its direct physical interpretation as applied force are valid in any frame.

The proper term for rate of change of velocity, as you note, is "coordinate acceleration", and as the name implies, it depends on your choice of coordinates, as of course does velocity itself.

 

[JohnNemo]

See here:

https://www.physicsforums.com/help/latexhelp/

Many thanks. I have managed to edit my former post and put the formula in.

 

[JohnNemo]

This is not correct. Special relativity is "special" because it only works if there is no gravity, i.e., if spacetime is flat. But in flat spacetime, SR can handle accelerating objects and accelerating frames just fine.

I'll take your word for it, but I'm puzzled as to where I got this idea from because I am sure I have read this lots of times. Is it common for it to be taught that "special relativity does not apply to accelerating frames" with the teacher eliding gravity and acceleration (and implicitly leaving of account the possibility of acceleration in flat spacetime due to e.g. electromagnetic force)?

Proper acceleration is not the rate of change of velocity. It is best thought of as applied force (and as you say, gravity is not a force in GR so it doesn't count here) divided by the object's mass. So when you move your phone around and the accelerometer number changes, it's not telling you how the phone's speed changes; it's telling you how much force you are applying to move the phone (and of course this is in addition to the force of the Earth pushing you up and transmitted through you to the phone).

It just so happens that, in a frame of reference fixed to the Earth, you can, with an appropriate choice of units, make the accelerometer number equal to the rate of change of velocity. But that correspondence is frame-dependent; in a different frame it won't be there. But the proper acceleration and its direct physical interpretation as applied force are valid in any frame.

The proper term for rate of change of velocity, as you note, is "coordinate acceleration", and as the name implies, it depends on your choice of coordinates, as of course does velocity itself.

This does come as a surprise - 'proper acceleration is not the rate of change of velocity'

Wikipedia says that

'In relativity theory, proper acceleration is ... acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured.'

https://en.wikipedia.org/wiki/Proper_acceleration

Is the Wikipedia definition problematic?

 

[PeterDonis]

I'm puzzled as to where I got this idea from because I am sure I have read this lots of times.

Yes, you might have. In the early days of SR, there was some confusion about this, so you can find sources from that time period (including IIRC some articles and letters by Einstein) that say it only works in inertial frames. You can also find more recent pop science sources, by people who have read those early writings and do not understand how SR has developed since that time, that make the same mistaken assertion. But yes, it is mistaken. Any modern textbook on relativity will tell you just what I said: that it can handle acceleration and accelerated frames just fine, as long as spacetime is flat.

Is it common for it to be taught that "special relativity does not apply to accelerating frames"

Not if the teacher is using a modern textbook. If they aren't, that's a problem with the teacher.

Is the Wikipedia definition problematic?

No, it's equivalent to the definition I gave. But it doesn't say what you think it says. It says that proper acceleration is equal to (coordinate) acceleration relative to an inertial observer who is momentarily at rest relative to the accelerating object. The "momentarily" is crucial: it means that which inertial observer it is changes from moment to moment along the trajectory of the accelerating object. So there is no way to define proper acceleration as "rate of change of velocity", because that would require it to be a rate of change with respect to a single observer. You can't define a rate if you constantly change what the rate is relative to.

 

[JohnNemo]

No, it's equivalent to the definition I gave. But it doesn't say what you think it says. It says that proper acceleration is equal to (coordinate) acceleration relative to an inertial observer who is momentarily at rest relative to the accelerating object. The "momentarily" is crucial: it means that which inertial observer it is changes from moment to moment along the trajectory of the accelerating object. So there is no way to define proper acceleration as "rate of change of velocity", because that would require it to be a rate of change with respect to a single observer. You can't define a rate if you constantly change what the rate is relative to.

When I read the Wikipedia article definition and saw ‘momentarilly’ I immediately thought ordinary differential calculus. A small slice of time yields a result approximating to the proper velocity and it tends to the exact value as the elapsed time of the slice of time tends to zero. Have I misunderstood?

 

[PeterDonis]

Have I misunderstood?

Yes. "Momentarily" in the Wikipedia passage in question means what I explained in my previous post.

 

[JohnNemo]

Yes. "Momentarily" in the Wikipedia passage in question means what I explained in my previous post.

There is a bit of context to the Wikipedia entry

Here https://en.wikipedia.org/wiki/Proper_acceleration

The full Wikipedia quotation is

‘In relativity theory, proper acceleration[1] is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured.’

And the second occurrence of the word ‘acceleration’ hyperlinks to https://en.m.wikipedia.org/wiki/Acceleration

which says

‘Acceleration, in physics, is the rate of change of velocity of an object with respect to time.’

That definition of acceleration is the one I am used to. I can understand that ‘proper acceleration’ has a special meaning but I have difficulty coming to terms with the idea that rate of change of velocity doesn’t come into the definition of ‘proper acceleration’ somewhere.

Can you help me to understand this?

 

[PeterDonis]

Can you help me to understand this?

The short answer is, Wikipedia is not an acceptable source.

The slightly longer answer is, you can't understand things by reading definitions. You have to actually think about the physics.

My previous posts in this thread have explained the distinction between coordinate acceleration and proper acceleration and why it's important, based on the physics, not definitions. You can define words however you want, although it certainly helps to understand the standard definitions used in physics textbooks and papers. The important thing is not the words but the physics.

I have difficulty coming to terms with the idea that rate of change of velocity doesn’t come into the definition of ‘proper acceleration’ somewhere.

I can understand that you might have difficulty, but I've already explained why "rate of change of velocity" won't work as a definition of proper acceleration. I'm not sure what I can add to what I've already said.

 

[PeterDonis]

I'm not sure what I can add to what I've already said.

Perhaps I can at least add an example. Suppose you are standing (or sitting, as I am while I type this) on the surface of the Earth. Relative to the Earth, the rate of change of your velocity is zero. But your proper acceleration is not; there is a nonzero force pushing up on you and causing you to feel weight. So obviously proper acceleration cannot be the same as rate of change of velocity.

But, you object, the "rate of change of velocity" definition (in Wikipedia, which, as I said, is not an acceptable source, but let that pass) said it was relative to an inertial observer. An observer at rest relative to the Earth's surface is not inertial. That's true; but as I pointed out, which inertial observer you have to use in this definition changes from moment to moment. Suppose, for example, that the Earth did not impede the motion of observers through it, so we could imagine a freely falling observer rising up towards you, standing on the Earth's surface, momentarily coming to rest relative to you, and then falling back down again. At the instant this observer was momentarily at rest relative to you, your rate of change of velocity, relative to him, would be equal to your proper acceleration, yes. But the next moment, that observer is no longer momentarily at rest relative to you; he is falling back down. And if you do the math, your rate of change of velocity relative to him will not remain constant; it will change. But your proper acceleration will remain the same.

You could also consider a whole family of such freely falling observers, each one momentarily coming to rest relative to you at a different instant. Your rate of change of velocity relative to each one, at the instant they are momentarily at rest relative to you, would be equal to your proper acceleration. But which observer that was would change from instant to instant. That means that, for your "rate of change of velocity" to remain constant--as it would have to if it is going to be equal to your proper acceleration--you would have to change which frame you used to evaluate it, from moment to moment. That's not a valid "rate of change of velocity". But it in no way changes the fact that your proper acceleration is constant: you feel it as a constant weight, and that's a physical fact that no amount of juggling with definitions or frames can change.

 

[JohnNemo]

Perhaps I can at least add an example. Suppose you are standing (or sitting, as I am while I type this) on the surface of the Earth. Relative to the Earth, the rate of change of your velocity is zero. But your proper acceleration is not; there is a nonzero force pushing up on you and causing you to feel weight. So obviously proper acceleration cannot be the same as rate of change of velocity.

But, you object, the "rate of change of velocity" definition (in Wikipedia, which, as I said, is not an acceptable source, but let that pass) said it was relative to an inertial observer. An observer at rest relative to the Earth's surface is not inertial. That's true; but as I pointed out, which inertial observer you have to use in this definition changes from moment to moment. Suppose, for example, that the Earth did not impede the motion of observers through it, so we could imagine a freely falling observer rising up towards you, standing on the Earth's surface, momentarily coming to rest relative to you, and then falling back down again. At the instant this observer was momentarily at rest relative to you, your rate of change of velocity, relative to him, would be equal to your proper acceleration, yes. But the next moment, that observer is no longer momentarily at rest relative to you; he is falling back down. And if you do the math, your rate of change of velocity relative to him will not remain constant; it will change. But your proper acceleration will remain the same.

You could also consider a whole family of such freely falling observers, each one momentarily coming to rest relative to you at a different instant. Your rate of change of velocity relative to each one, at the instant they are momentarily at rest relative to you, would be equal to your proper acceleration. But which observer that was would change from instant to instant. That means that, for your "rate of change of velocity" to remain constant--as it would have to if it is going to be equal to your proper acceleration--you would have to change which frame you used to evaluate it, from moment to moment. That's not a valid "rate of change of velocity". But it in no way changes the fact that your proper acceleration is constant: you feel it as a constant weight, and that's a physical fact that no amount of juggling with definitions or frames can change.

I get your point that logically it is not valid to measure something from a constantly changing point of reference. But I suppose one difficulty I have is that proper acceleration is measured in distance divided by the square of time so it almost looks like the idea of it being a rate of change of velocity (distance divided by time) is ‘built in’ to the nature of the quantity being measured.

 

[PeterDonis]

proper acceleration is measured in distance divided by the square of time

The fact that a quantity has units of distance divided by time squared does not mean it must have a physical interpretation as a rate of change of velocity. Distance divided by the square of time is also force divided by mass. So the units of proper acceleration are equally consistent with a physical interpretation as force divided by mass. To see which of these interpretations makes more sense, you have to look at the actual physics.

 

[JohnNemo]

The fact that a quantity has units of distance divided by time squared does not mean it must have a physical interpretation as a rate of change of velocity. Distance divided by the square of time is also force divided by mass. So the units of proper acceleration are equally consistent with a physical interpretation as force divided by mass. To see which of these interpretations makes more sense, you have to look at the actual physics.

In my write-up, which you were kind enough to comment on, I said

’It is the same when you are standing on the Earth, the force of the Earth is pushing you in an upwards direction and causing you to accelerate at 9.8 metres per second per second. This acceleration, measured against spacetime (which is a real physical thing, remember) is called proper acceleration. You can measure proper acceleration using an instrument called an accelerometer.’

Is that a valid way of expressing it?

 

[PeterDonis]

Is that a valid way of expressing it?

Only if you understand that "accelerate at 9.8 meters per second per second" does not mean that this quantity is a rate of change of velocity. But of course that wording is going to strongly invite that interpretation. So I would not recommend it. I would recommend focusing on the fact that you feel weight when the Earth pushes on you.

 

[Dale]

‘In relativity theory, proper acceleration[1] is the physical acceleration (i.e., measurable acceleration as by an accelerometer)

Physics is a science, which means that it can be investigated using the scientific method. So the most important way to define something in physics is in terms of the measurements that can be performed to quantify it. So I would take the parenthetical comment as the key definition. Proper acceleration is the thing measured by an accelerometer, full stop.

You can set up local coordinate systems such that the second derivative of position is equal to the proper acceleration for any object whose first derivative is momentarily zero, and such coordinates are known as inertial coordinates, but that is a property of the inertial coordinates. The proper acceleration is not defined in that way, it is more directly defined than either position or velocity. This makes sense because both position and velocity are relative quantities, whereas proper acceleration is invariant.

 

[JohnNemo]

@JohnNemo overall this looks like a good summary. There are just a couple of points that need correction:
This is not correct. Special relativity is "special" because it only works if there is no gravity, i.e., if spacetime is flat. But in flat spacetime, SR can handle accelerating objects and accelerating frames just fine.
Proper acceleration is not the rate of change of velocity. It is best thought of as applied force (and as you say, gravity is not a force in GR so it doesn't count here) divided by the object's mass. So when you move your phone around and the accelerometer number changes, it's not telling you how the phone's speed changes; it's telling you how much force you are applying to move the phone (and of course this is in addition to the force of the Earth pushing you up and transmitted through you to the phone).

It just so happens that, in a frame of reference fixed to the Earth, you can, with an appropriate choice of units, make the accelerometer number equal to the rate of change of velocity. But that correspondence is frame-dependent; in a different frame it won't be there. But the proper acceleration and its direct physical interpretation as applied force are valid in any frame.

The proper term for rate of change of velocity, as you note, is "coordinate acceleration", and as the name implies, it depends on your choice of coordinates, as of course does velocity itself.

I have made changes to those two areas. Does it look OK now?



What is relative in General Relativity?The theory of Special Relativity is fascinating. To get a basic understanding of it you only need a good book, basic algebra, and a willingness to have your intuitive ideas about time, space and motion upset.

Special relativity is built on three ideas

· Speed is relative

· Except the speed of light which is everywhere the same

· Nothing can travel faster than the speed of light

If you are walking at about 3 mph inside a railway carriage, walking towards the front, but the train in traveling at 100 mph, we might be inclined to say that your ‘real’ or ‘absolute’ speed is about 103 mph, but actually all we can say is that your speed is about 103 mph relative to the Earth. If we chose to measure your speed relative to the Sun it would be different again, or if it is measured relative to a distant galaxy it will be different again. It turns out that there is no such thing as ‘real’ or ‘absolute’ speed: you can only measure the speed of an object relative to some other object.

The speed of light is 671 million miles per hour. Suppose a spaceship is traveling at 400 million miles per hour away from the Earth. Some time ago the spacecraft launched a smaller craft which has picked up speed and is now traveling at 350 miles million miles per hour relative to (and in front of) the mothership. We would expect that the smaller craft would be traveling at 750 million miles per hour relative to the Earth, but it turns out that that is not the case: It cannot be the case because nothing can travel faster than the speed of light.

If B is traveling at speed S relative to A, and C is traveling at speed T relative to B (in the same direction) then we expect the speed of C relative to A to be S + T. But it turns out that this is not correct and C’s speed relative to A is actually

$$\frac {S+T} {1+ ST/c^2}$$where c is the speed of light.

If we do the mathematics it turns out that the smaller craft is traveling at 553 million miles per hour relative to the Earth, still less than the speed of light.

This is a strange result and there are other strange results of the theory of Special Relativity. It turns out that, measured relative to the Earth, the spaceship is shorter than it was when it was at rest on the Earth. This phenomenon is known as the Lorentz contraction. The crew of the spaceship do not notice any difference – from their frame of reference the length of the spaceship is the same as it has always been, but, when measured from the Earth, it is shorter.

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.General Relativity

Special Relativity is called ‘special’ because it applies to the ‘special case’ of objects which are not subject to gravity - e.g. in deep space a long way from the nearest star. It works to a good approximation in weak gravity but for any situation where gravity is significant, you need General Relativity. If you have studied Special Relativity and are now about to look at General Relativity, you might assume from the name that General Relativity must be based on the idea that everything, including acceleration, is relative. But this is not the case and the name General Relativity is potentially misleading. It is important, when considering General Relativity, to get clear in your mind what is, and is not, relative.

But, first of all, let us talk about space-time. Before Einstein, there was some debate about what space is – is it a real thing or is it just the absence of anything. In General Relativity space, or rather space-time (four dimensions including time) is a real physical thing. You can imagine it as a grid – a grid which is distorted where there is mass and/or energy. It is distorted most next to large masses and if a mass is accelerating that adds to the distortion. Now, in General Relativity, gravity is not a ‘force’ but is explained by the distortion of spacetime. There are natural paths in spacetime which any object which is not being pushed or pulled by any force will follow – these are called geodesics. The presence of mass distorts spacetime so that geodesics tend to curve towards the mass. Thus the reason why objects tend to move towards mass is not because of some force emanating from the mass but because spacetime has been curved by the mass.

So if gravity is not a force, why does it feel to us like a force? Imagine that someone lifts up a coin and drops it. When the coin is in mid air, moving towards the ground, it is moving on a geodesic taking it towards the centre of the Earth, but when the coin hits the ground, the force from the ground prevents it moving any further towards the centre of the Earth. The force which we think of as gravity is actually not a force pulling us down but a force pushing us up!

Here is another illustration. Suppose you are in a spaceship somewhere in deep space a long way from the nearest star, just drifting because the engines are switched off. You switch on the engines and the spaceship starts accelerating at, say, 1g. You feel yourself being pulled back against the cabin wall/floor towards the rear of the spaceship, but although it feels like that, you are not actually being pulled back at all: you are being pushed forwards by the force of the cabin wall/floor which is (together with the rest of the spaceship) accelerating forwards.

It is the same when you are standing on the Earth, the force of the Earth is pushing you in an upwards direction and causing you to accelerate at 1g. This acceleration, measured against spacetime (which is a real physical thing, remember) is called proper acceleration. You can measure proper acceleration using an instrument called an accelerometer.

You almost certainly already possesses an accelerometer because there will be one inside your mobile phone. Your phone uses it to, for example, turn the display to landscape as you rotate the phone. You can download an accelerometer app which will actually display the proper acceleration. When the phone is lying on your desk it will show an acceleration of 1g. If you took it into a rocket and blasted off, of course it would show a higher reading.

In General Relativity you can choose any reference frame (including a rotating frame) and measure a body’s acceleration from that frame. This measurement of acceleration is called co-ordinate acceleration. The co-ordinate acceleration of a body can be different when measured from different reference frames. However it is important to realize that, irrespective of the reference frame, the proper acceleration of a body is invariant. Whatever reference frame you are in you can get out your binoculars and look at the reading on an accelerometer on that body and it will be whatever it is.

It is important to get this clear because if you have studied Special Relativity you will have made the mental leap from thinking about velocity as being absolute to realising that velocity is entirely a matter of velocity relative to a reference frame, and you might assume that in General Relativity acceleration is entirely a matter relative to a reference frame, but this is not the case. Proper acceleration is invariant because it is measured against the local spacetime geometry. There is no equivalent for velocity because the geometry of spacetime does not allow velocity itself to be measured against it.

If you have not come across the phrase proper velocity, you can skip this paragraph. If you have come across the idea of proper velocity you may be thinking that this is the equivalent – for velocity – of proper acceleration, but this is not really so: Proper velocity relative to an observer divides observer-measured distance by the time elapsed on the clocks of the traveling object, so it is still a relative measurement and is not (unlike proper acceleration) invariant.Rotation

Rotation does not have a well defined meaning in General Relativity. Part of the reason for this appears to be that if rotation is traditionally thought of as acceleration towards an axis coupled with velocity perpendicular to the direction of acceleration, it consists of a mixture of invariant and relative elements. The best I have been able to understand how General Relativity treats rotation is that there are four invariant indicators (which, strangely, are not necessarily all present together) which roughly equate to ‘rotation’. They are (1) pattern of proper acceleration; (2) precession; (3) Sagnac effect; (4) vorticity.Ptolemy and Copernicus

In the Middle Ages it was thought, following Ptolemy, that the Earth was fixed immovable at the centre of the universe, and the Sun orbited it. Then, at the end of the Middle Ages Copernicus, proposed that, on the contrary, the Sun was fixed immoveable at the centre of the universe and the Earth orbited the Sun. Now that we are starting to learn about Relativity it is a fascinating exercise to reassess this controversy and see who (if anyone) was right. If this does not interest you, you can stop reading now – you can learn General Relativity perfectly well without knowing anything about this historical controversy, but, if you are interested, looking at this controversy will help to apply and consolidate some of the basic features of General Relativity as discussed above.

First a reminder about what the controversy was about. Copernicus showed that you can model the motions of planets in relation to the Sun and that this is much simpler than modelling them in relation to the Earth. This insight was generally welcomed as useful irrespective of whether the Earth actually moved. For example, Tycho Brahe (1546 to 1601) combined belief in the immovability of the Earth with use of Copernicus’ calculations. When Copernicus’ book De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) was published, in 1543, the book started with an unattributed letter actually written by the Lutheran preacher Andreas Osiander who had been responsible for supervising the printing and publication. This letter, whose inclusion in the book was probably not authorised by Copernicus, was clearly designed to emphasise the uncontroversial mathematics, and deflect criticism of the controversial matter of whether the Sun or the Earth moves (Copernicus’ views on the latter had already received criticism from the Lutheran leaders, Philip Melanchthon and Martin Luther himself).

“There have already been widespread reports about the novel hypotheses of this work, which declares that the Earth moves whereas the sun is at rest in the centre of the universe Hence certain scholars, I have no doubt, are deeply offended and believe that the liberal arts, which were established long ago on a sound basis, should not be thrown into confusion. But if these men are willing to examine the matter closely, they will find that the author of this work has done nothing blameworthy. For it is the duty of an astronomer to compose the history of the celestial motions through careful and expert study. Then he must conceive and devise the causes of these motions or hypotheses about them. Since he cannot in any way attain to the true causes, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry for the future as well as for the past. The present author has performed both these duties excellently. For these hypotheses need not be true nor even probable. On the contrary, if they provide a calculus consistent with the observations, that alone is enough...”And Copernicus himself, in De revolutionibus, addresses the controversy head on. He starts with the arguments against the Earth moving relied on by ancient philosophers such as Aristotle and Ptolemy. Everything is drawn towards the centre of the Earth and would be at rest at the centre – if not checked by the surface of the Earth. Therefore the entire Earth is at rest. The Earth is heavy and not apt to move. If such a heavy object did move its motion would be violent. Copernicus deals with these arguments and then adds some arguments in favour of the Earth’s movement – e.g. the Earth is a sphere and it is natural for spheres to move in a circle.For Copernicus the controversy is over whether the Earth moves or the Sun moves – the possibility that both move being discounted:

“Hence I feel no shame in asserting that this whole region engirdled by the moon, and the centre of the earth, traverse this grand circle amid the rest of the planets in an annual revolution around the sun. Near the sun is the centre of the universe. Moreover, since the sun remains stationary, whatever appears as a motion of the sun is really due rather to the motion of the earth...

All these statements are difficult and almost inconceivable, being of course opposed to the beliefs of many people. Yet, as we proceed, with God’s help I shall make them clearer than sunlight, at any rate to those who are not unacquainted with the science of astronomy...”

So what does General Relativity tell us about who was right? Does the Earth move or does the Sun move?

The first thing to say is that the fact that the mathematics of General Relativity allows you to choose any frame of reference, including a rotating frame, including the frame of the Sun or the frame of the Earth, and describe phenomena as measured from that reference frame, is a bit of a red herring. That just shows that the mathematics is very versatile, but it does not signify anything about the physics of General Relativity (just as Copernicus’ calculations did not, of themselves, prove the matter one way or another).

On a large scale the matter and energy in the universe is isotropic so that if you have a region of space some distance away from the nearest star, such as our solar system, the matter and energy about that region is spherically symmetric, and the spacetime in that region would be ‘flat’ if it were empty. Consequently the spacetime geometry of the region is entirely defined by its contents - our Sun and the planets, and because over 99 per cent of the matter and energy is contained in the Sun, the spacetime geometry will, to a good approximation, be the geometry of a single source of gravity, the Sun, with all the planets moving on geodesics. A consequence of this is that the movement of the planets in a reference frame which is non-rotating relative to the distant stars and in which the Sun is at rest, is a much more regular movement (almost circular) than the movement of the planets in any reference frame in which the Earth is at rest. But this just means that the mathematics is simper: it does not help us decide which one is actually moving.

What about the invariant indicators of rotation referred to earlier? After all a body which is orbiting should exhibit at least some of these. Can we use these to determine whether the Sun orbits the Earth or the Earth orbits the Sun? Apparently not because these invariant indicators of rotation are present in both the Sun and the Earth.

So we have to conclude that Copernicus and Ptolemy were both wrong (or both half right depending how you look at it).

The final word goes to Einstein:

"Can we formulate physical laws so that they are valid for all CS (=coordinate systems), not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any CS. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS could be used with equal justification. The two sentences, 'the sun is at rest and the Earth moves', or 'the sun moves and the Earth is at rest', would simply mean two different conventions concerning two different CS. Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!" (The Evolution of Physics, 1938)

 

[PeterDonis]

@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?

 

[pervect]

I'll take your word for it, but I'm puzzled as to where I got this idea from because I am sure I have read this lots of times. Is it common for it to be taught that "special relativity does not apply to accelerating frames" with the teacher eliding gravity and acceleration (and implicitly leaving of account the possibility of acceleration in flat spacetime due to e.g. electromagnetic force)?

In introductory courses, special relativity is usually taught in a manner that requires one to use inertial frames of reference and cartesian coordinates . Much the same is true for Newtonian mechanics, the math is much simpler to learn. Sometimes one might introduce simple alternative coordinates such as polar coordinates, but the methods needed to use general coordinates are not taught at introductory levels.

In general, though, the coordinate system doesn't matter to the theory. There is nothing physical about a choice of coordinate system, though people sometimes falsely assume that there is because they have only been taught how to work with the theory using specific conventions and coordinate choices.

For people familiar with the rather sophisticated methods needed to use generalized coordinates, the theory of special relativity is applicable to any flat space-time. Accelerated frames of reference can be regarded as a specific application of generalized coordinates. So can rotating frames of reference.

One route to coordinate independence is through Lagrangian mechanics. Rather than expressing non-relativistic mechanics by Newton's laws, one can express it via the principle of least action.

Goldstein's "Classical Mechanics" is a standard textbook that explains the Lagrangian formulation in detail, and it presents both the classical and the relativistic Lagrangian formulations.

People not familiar with these methods spend a great deal of time worrying about problems that can be solved by applying them. Most likely they could spend their time more productively by learning the advanced methods, but this does require work, effort, dedication, and a lack of math phobia - so it might not be as enjoyable. This is especially true for people cursed with math phobia :(.

Once one has such a Lagrangian formulation of physics, going to special relativity is basically just a matter of using a different Lagrangian - a different action. The principle that the action is least is independent of the choice of coordinates, one only needs the correct formula for the action for the specific coordinates one adopts to change coordinates.

Much the same can be said for rotating coordinates, the same generalized coordinate techniques can be used. Defining the rotating coordinates is a little tricky, though. The issues involve defining a notion of "now" that's applicable to the rotating coordinate system. Some of the common techniques used in non-rotating coordinate systems needed a bit of modification.

In a non-rotating coordinate system, all clocks run at the same rate, and can be synchronized via the Einstein convention, which yields a well-defined and standardized method of choosing coordinates. People do still stumble over the fact that choosing a different inertial frame of reference in special relativity implies a different clock synchronization convention. In a rotating coordiante system, all clocks do not run at the same rate, in the first place, so Einstein's convetion isn't directly applicable. Furthtermore, even if one chooses a subset of clocks that do run at the same rate (clocks at the same radius from the origin of the rotating coordinate system), there are issues such as the Sagnac effect that still make the Einstein method not work. There are known methods for defining time in a rotating coordinate system, of course, the most obvious (and commonly used) one is to setup a non-rotating coordinate frame of reference and use it's defintion of time.

One can turn this around, and make the point that non-rotating coordinate systems in flat space-time are defined by the ability to set up a frame of clocks that all run at the same rate and which can all be synchronized by the Einstien convention (and remain synchronized because they run at the same rate). Then one has a coordinate independent definition of what it means to have a non-rotating frame of reference.

 

[JohnNemo]

@JohnNemo the corrections you made look fine

Thank you for checking that. I would expand the section on time dilation to mention redshifting like 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.

Because the ground crew and the spaceship are traveling away from each other, light traveling between them will be redshifted and this will create the appearance that the ground crew are in slow motion as seen from the spaceship. However it is important to realize that even after the crew on the spaceship take account of the redshifting effect in their calculations, they will still, even after taking that into account, measure time passing more slowly for the ground crew. Likewise the ground crew will measure time passing more slowly on the spaceship even after taking account of the redshift effect."

I hope this is also correct

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

 

[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.