Reference Frames

[Farsight]

REFERENCE FRAMES

 

You know I was talking about magnetic fields and electric fields? When you move through an electric field you see it as a magnetic field. That’s Relativity. It’s the same old thing really. The difference is down to you. Sometimes you don’t realise that things are the same because you see them a particular way. Because you walk around all your life wearing some very special sunglasses. They’re like Ray-Bans. You grow up with them, so much so that you don’t know you’re wearing them.

 

 

They colour your vision but you cannot see how. They stop you seeing the light for what it really is. I need to talk about them, because I want to get to the bottom of Special Relativity and talk about time, and I want to get to the bottom of General Relativity and talk about gravity. But the things that colour your vision aren’t sunglasses. What they are is reference frames.

 

Let’s have a little gedanken, a thought experiment. It involves my spaceship. Remember my spaceship?

 

 

You’re the copilot, and I’ve just taken you down to the cargo bay and shown you “the box”. It’s a ten foot canister sitting in front of the airlock. It’s comfortable, cushioned, equipped with an air supply so you don’t need a spacesuit, with thruster rockets for positioning and trim and emergency escapes. And as you may have gathered, you have to get in the box.

 

You climb in, find the light switch and turn it on, then I swing the hatch closed and give you a slam slam goodbye. You hear me leave and the pumps evacuating the air, followed by the grinding of steel as the outer doors open. There’s a jolt as the launch ram pushes you outside the ship, and then everything goes quiet. There are no windows in this box. There is however a radio so we can stay in touch. I call you up to say I’m pulling back a little, leaving you there in your box.

 

 

You say A-OK catch you later, and enjoy a few zero-gravity games, doing somersaults and pushing yourself from one side of the box to the other practising your racing turns. After a while you find the tennis ball in your jumpsuit pocket and throw it against the opposite wall, smiling when it bounces straight back into your waiting hand. The radio bleeps into life and it’s me, asking how you’re doing. Fine you reply, I ask if you can detect any acceleration and you say Nope. You’re cool, because you’re weightless, it’s fun. You're in the box. There are no windows. You chuck the ball across the inside of the box and it goes straight as a die, bounces off the side, and back into your hand. You are in an inertial reference frame. You can feel no gravitational force, and you can detect no gravitational force acting upon the ball. You are not accelerating.

 

I ask if you’d like to try a little acceleration, you say Sure! and then you feel and hear the thruster rockets burning. You find yourself pressed back, and now one side of the box feels like it’s the floor. You stand up and it’s like being on the surface of the moon, or shipside under artificial gravity. You can feel your weight. You throw the ball and it travels in a lazy arc towards the floor and bounces around a little before settling there. You are now in a non-inertial reference frame, and you know you’re accelerating. You can feel it, you can measure it locally, within your frame, within your box.

 

 

I shut off the thrusters so you’re back weightless again, floating in your inertial reference frame. You can no longer feel any acceleration. You can’t measure any. The ball flies straight as a die. As far as you are concerned, you are not accelerating.

 

Did I mention that you’re falling into a black hole? No? Oops. Sorry about that.

 

Here’s the deal: a body in freefall is not accelerating. If you say it is, you're mixing reference frames. You're looking at yourself from some reference frame out somewhere in space rather than from your reference frame, that of the body in freefall. Don’t be mistaken about this. The Principle of Equivalence between gravity and an accelerating box applies when you're standing on a planet, not when you're in freefall.

 

 

There really is no force acting on your freefalling body. You can't feel any, you can't see any, you can't measure any. Because there is none. If you looked out of a window to see a moon going backwards, you're mixing frames. This is why gravity is not technically a force. It exerts no force on you. In your reference frame you are not accelerating. You never can be, because in your reference frame your velocity must be zero. If you say it isn’t, you’re mixing frames. You really really are not accelerating, and that’s why you can throw that ball straight. And that’s why Einstein called gravity a pseudoforce.

 

Meanwhile I’m sitting pretty, out in space in the ship where the black hole gravity is so neglible that I can also consider myself to be in an inertial reference frame. All I have to do is switch off the ship gravity, and I can play ball games just like you. But when I look through the viewscreen I see you falling faster and faster towards the black hole, knowing that as far as you’re concerned, you’re in an inertial reference frame just like me. We’re both in inertial reference frames, but yours isn’t inertial as far as I’m concerned, and vice versa. What’s happening is that your inertial reference frame is changing and you can’t see it.

 

But you can detect something, if you have the right equipment. And that you do. Because the radio crackles and it’s me again, telling you open a concealed hatch and pull out an apparatus that looks like a long dumbell. I tell you to perform a "Pound-Rebka" experiment, and you follow my instructions and find that you get a photon blueshift reading when the instrument is pointed in a given direction. I tell you it’s pointed at the black hole, and what you’re measuring is a slight tidal force in that direction.

 

 

Note that a “proper” gravitational field is never uniform. Sometimes the tidal force is neglible, but it is never ever zero. If gravitational felds were uniform, they’d be hills without slopes, they’d be flat, so they wouldn’t be hills at all. The tidal force is always there because of a slight change in the gravity in a given direction. This is where the Principle of Equivalence isn’t quite perfect, because there is no tidal force in a simple accelerating box in free space. Einstein knew this, but people only look at what he generalised in 1911 and take it all too far. Here’s a quote from a guy called John R Ray dating from 1977:

 

The first thing to note about the 1911 version of the principle of equivalence is that what in 1911 is called a uniform gravitational field ends up in general relativity not to be a gravitational field at all – The Riemann tensor is here identically zero. Real gravitational fields are not uniform since they must fall off as once recedes from gravitating matter.

 

It’s really obvious when you think about it. But people never do. They never actually look at the reference frames that they look through. You’re in your box, and "gravity" is continuously changing your inertial reference frame. But you’re in it, you are immersed in your reference frame, you can’t see it, it’s what you look through to see the world, you can’t see it changing. All your photons are blueshifting, but you don’t notice it. Your time is dilating, but you don’t notice it. Your seconds are changing, and your metres too, but not to you, because you can’t see it.

 

Your reference frame is how you see the world, but it isn’t how the world is. Because in your reference frame, your velocity is always zero. And that simple fact means you are at odds with Copernicus, because in your reference frame, the sun goes round the earth.

 

 

I don’t like reference frames. They aren’t real, they don’t actually exist, they get in the way instead of making things clearer. People talk about them too much, more than Einstein did. Let’s use them less, and learn to look at the world the way it really is.

 

Is that the time? It’s time for tea. We will continue this gedanken another time. OK, thrusters on, full boost. Come on home, back to the ship. Time is of the essence.

[Farsight]

I've been writing further essays which I intend as chapters of a popular science book. This one started life as a mention within GRAVITY EXPLAINED, and grew via a thought experiment I was talking to Zanket about. I then expanded it within a chapter on black holes, and have now juggled things around and made it a little lead-in chapter all on its own.

 

I'd be grateful if anybody could point out any errors herein, and otherwise I hope it proves useful.

[sanctus]

I have to say it was a very interesting read Popular, but there is one thing missing an alternative...I really wonder how you want to explains things without reference frames? Isn't it enough, as you point out very well, to know that all is relative to the considered reference frames and formulate everything in a covariant form (so that no new things happen when you change reference frame)?

[Farsight]

I'm not saying we shouldn't use them at all, sanctus, just less. And we need to remember what it is that we're talking about.

 

For example, imagine a hypothetical charged particle, a "point particle", just sitting there doing nothing. It has an electrostatic field extending outwards. It has no magnetic field. But if you zip past it, or it zips past you, it does. It had an electromagnetic field all along, which you view an an electric field and/or a magnetic field. How you see it depends on your relative motion, and your reference frame compared to it. But the particle hasn't changed at all.

 

It's similar with length contraction. If you travel at .99c you will measure all distances as if they're contracted sevenfold in the direction of travel. You would measure the entire universe as being contracted sevenfold in the direction of your travel. But the universe hasn't changed one iota.

 

All this ties in with the "ontological viewpoint" that I use in my various essays.

[Qfwfq]

Here’s the deal: a body in freefall is not accelerating. If you say it is, you're mixing reference frames.

I'd say that you are choosing a reference frame.

 

The Principle of Equivalence between gravity and an accelerating box applies when you're standing on a planet, not when you're in freefall.

It's always valid. In a differential sense, of course. When in free fall, you simply "don't need" it because you're "already" in free fall. But then, you can use it the other way around, to say that the falling ball simply "isn't being accelerated upward" and that is where gravity becomes a pseudo-force, or apparent force, exactly like all inertial forces.

 

Newton stated it only for a uniform gravitational field (with a remark about approximation for nearly uniform cases) but GR states in in general, in differential terms. Before SR and Minkowskian geometry, all attempts to do this had failed.

 

In your reference frame you are not accelerating. You never can be, because in your reference frame your velocity must be zero.

True, this is so even when the thrusters are going. That's why you say that the ball is accelerating, instead of the spaceship.

 

How sure is John R Ray that the Riemann tensor in a uniform gravitational field is identically zero? The spatial submanifold is flat, but space-time isn't.

 

I don’t like reference frames. They aren’t real, they don’t actually exist, they get in the way instead of making things clearer. People talk about them too much, more than Einstein did. Let’s use them less, and learn to look at the world the way it really is.

And how is the world, really? The whole point of the principle of relativity is reference frames. They were the whole point that Einstein was addressing.

[Farsight]

Thanks for the feedback, Qfwfq.

 

I'd say that you are choosing a reference frame.

 

Noted. I'll see about working that in.

 

It's always valid. In a differential sense, of course. When in free fall, you simply "don't need" it because you're "already" in free fall. But then, you can use it the other way around, to say that the falling ball simply "isn't being accelerated upward" and that is where gravity becomes a pseudo-force, or apparent force, exactly like all inertial forces.

 

Hmmn. I wrote what I did because I'd been talking to somebody who compared the freefall acceleration to the acceleration in a rocket-powered box. I'll check my wording to make it clearer.

 

Newton stated it only for a uniform gravitational field (with a remark about approximation for nearly uniform cases) but GR states in in general, in differential terms. Before SR and Minkowskian geometry, all attempts to do this had failed.

Please can you clarify this?

 

How sure is John R Ray that the Riemann tensor in a uniform gravitational field is identically zero? The spatial submanifold is flat, but space-time isn't.

 

I can't speak for him. But I will say this: there is no such thing as a uniform gravitational field. It's a contradiction in terms. Like a flat hill. And I would hazard a guess that spacetime curvature is something that you take for granted and can not actually explain. I could talk at length about this, and it might take us some way off this topic, so I'll leave it there. I came across the quote by Ray in an interesting paper Einstein’s Gravitational Field by Pete Brown. See page 20. There’s also a quote by Synge:

 

http://xxx.lanl.gov/ftp/physics/papers/0204/0204044.pdf

 

“... I have never been able to understand this principle…Does it mean that the

effects of a gravitational field are indistinguishable from the effects of an

observer’s acceleration? If so, it is false. In Einstein’s theory, either there is

a gravitational field or there is none, according as the Riemann tensor does

not or does vanish. This is an absolute property; it has nothing to do with any

observers world line … The Principle of Equivalence performed the essential

office of midwife at the birth of general relativity, but, as Einstein remarked,

the infant would never have gone beyond its long clothes had it not been for

Minkowski’s concept [of space-time geometry]. I suggest that the midwife be

buried with appropriate honours and the facts of absolute space-time faced.”

 

And how is the world, really? The whole point of the principle of relativity is reference frames. They were the whole point that Einstein was addressing.

It's pure marble. If you catch my drift.

[snoopy]

Yeah popular,

 

I think I have to agree with the moderators on this one,

 

You need to pick a frame of reference its absolutely fundamental to living in a relativistic universe.

 

Newton had thought about relativity then rejected it on the grounds of points similar to the ones you are making but as it turns out Newton was wrong.

 

You need to pick a frame of reference to make relativity work and in the elevator/space rocket thought experiments Einstein was trying to say forces that are similar are actually the same.

 

It doesn't matter which frame of reference you pick you could pick the earth and have the entire universe moving relative to it. Or pick the rest of the universe and have the earth move relative to it.

 

But you must pick one, relativity doesn't work if you don't.

[jungjedi]

is that spaceship the serenity from firefly

Serenity (Firefly vessel - Wikipedia, the free encyclopedia)

[Farsight]

Noted snoopy. I'll think carefully about what you said.

 

jungjedi: yep. I've used it before when talking about "my spaceship" to illustrate a point.

[Qfwfq]

OK Popular, I needed to shake a few cobwebs out of my differential geometry, over the weekend I gave a quick read through a translation of Die Grundlagen der allgemeiner Relativitätstheorie. In a uniform field the intrinsic curvature is zero, only the embedding into [math]\norm\mathbb{R}^4[/math] is curved. I was somewhat confusing things. Actually, :doh: the doubt had already crossed my mind, due exactly to the trivial argument Einstein gave, it's the very first remark he makes after introducing the Riemann tensor with four indices. All the same, I don't get the drift about the equivalence principle.

 

Now the principle is obvious enough for a uniform field; in Newton's Principia it is one of the corollaries (I think VI but I can't remember for sure) after the three axioms. Einstein's Ansatz is that, given a point, even if the principle can't be applied in a finite region, there is a coordinate choice which is locally inertial for that point, meaning that SR is valid to the first order in a neighborhood of it. There is absolutely nothing wrong with this principle (except that it would obviously not apply at a singularity such as that of the Schwarzschild solution).

 

Before Einstein, some had thought of using the same idea in a 4-D euclidean space-time to describe gravity but attempts failed. I read somewhere that someone eventually proved that it can't be done, so it's only with Minkowski's metric that the idea works and not with Euclid's.

 

It's pure marble. If you catch my drift.

No, I don't catch it, sorry.

[Farsight]

OK Popular, I needed to shake a few cobwebs out of my differential geometry, over the weekend I gave a quick read through a translation of Die Grundlagen der allgemeiner Relativitätstheorie. In a uniform field the intrinsic curvature is zero, only the embedding into [math]\norm\mathbb{R}^4[/math] is curved. I was somewhat confusing things. Actually, :doh: the doubt had already crossed my mind, due exactly to the trivial argument Einstein gave, it's the very first remark he makes after introducing the Riemann tensor with four indices. All the same, I don't get the drift about the equivalence principle.

 

You might not like me for saying this but: there is no uniform field. A uniform field is a flat hill. And if it's flat there ain't no hill. It's a contradiction in terms. If it's locally flat your locality is of zero extent, and that's an abstraction that just isn't real. See my little lead-in essay on Reference Frames. Something is changing all the while, even across the extent of your local frame. It's "negligible" but not zero. If it was zero and your local frame was totally flat, things wouldn't fall down. The equivalence principle is OK up to a point, in that if you're standing on a planet it's like you're accelerating in a rocket-propelled box. But not exactly. You can measure the gravity gradient even locally via a Pound-Rebka experiment, and you just can't do this in a rocket-propelled box. Einstein knew this. See Pete Brown's paper Einstein's Gravitational Field on http://www.geocities.com/physics_world/ for an interesting read. He's pmb on Physics Forums. Good bloke.

 

Now the principle is obvious enough for a uniform field; in Newton's Principia it is one of the corollaries (I think VI but I can't remember for sure) after the three axioms. Einstein's Ansatz is that, given a point, even if the principle can't be applied in a finite region, there is a coordinate choice which is locally inertial for that point, meaning that SR is valid to the first order in a neighborhood of it. There is absolutely nothing wrong with this principle (except that it would obviously not apply at a singularity such as that of the Schwarzschild solution).

See above. I've read what I think is Einstein's original GR (darn, I can't find it, it wasn't this Einstein, Albert. 1920. Relativity: The Special and General Theory) and it didn't say anything about curvature. Nothing at all. Just accleration. Your neighbourhood is of zero size, and that means it doesn't actually exist. It's a locally flat portion of a gradient, and it's an abstraction that obscures what gravity actually is.

 

Einstein, some had thought of using the same idea in a 4-D euclidean space-time to describe gravity but attempts failed. I read somewhere that someone eventually proved that it can't be done, so it's only with Minkowski's metric that the idea works and not with Euclid's.

I'm an advocate of relativity. The essays I've written are grouped under a thread called RELATIVITY+. But I think Minkowski was plumb wrong. Einstein said "time is suspect", and I agree with him. We think of spacetime as something four dimensional, but there's only actually three dimensions of space plus another time "dimension" that is a relative measure of motion through space. It means time is just a measure of motion, and you can't move through it, so spacetime is an abstraction, which means curved spacetime is an abstraction too. Try explaining what it is and you'll see what I mean. See TIME EXPLAINED for further details.

 

No, I don't catch it, sorry.

It's what Einstein was working on. His "pure geometrical marble" is how you make matter out of space. I've got the recipe.

[Qfwfq]

I'm afraid we are seriously misunderstanding each other. I no longer get whether your aim is that of writing a divulgative explanation of what Einstein did say, or of promoting ideas of your own. Call a spade a spade.

 

Now what I know to be Einstein's original GR is what I cited yesterday (DGDAR). I can give you the full reference tomorrow, I don't have it with me, though I suspect you'd want an English translation of it anyway and you can prolly find a good one. In any case it isn't the best thing for learning GR, it isn't the most modern, nor mathematically best, treatment but it's the thing to refer to for talking about "the original". An example of a good course (and in English) is contained within Weinberg's Gravitation and Cosmology, a classic textbook. Einstein in DGDAR did not always use the exact same terms as those currently used but the intended meanings are the same. This is often a problem with historic analysis, when going back to look at what the chap actually said one must take the slightly different language into account.

 

Now this weekend I didn't even notice whether or not Einstein mentions the word curvature but this is irrelevant, what the Riemann tensor describes is intrinsic curvature. Now my cobwebs were exactly on the point that this is not acceleration (which instead is the curvature of the mapping) the intrinsic curvature is necessary due to the field not being uniform. This is obvious, and probably was when I had taken the course and passed the exam, but lately it was a bit less obvious... before I gave the matter a review. The fact that gravitational fields due to massive bodies aren't uniform does not defeat Einstein's argument, if you understand it properly.

 

Now for a locally inertial (terms that Einstein did not use in DGDAR, but very handy terms that were coined later to mean the exact same thing that he said) coordinate map, the changes in the neighbourhood of the given point are not "negligible" at all but instead infinitesimal of at least second order. A totally different concept altogether. The second order terms become negligible asymptotically with the distance of the given point from the massive body, they aren't negligible as you come toward small r values.

 

How, then, would a uniform gravitational field be represented in GR? Is the Riemann tensor the only one in GR? What is the description from the rocket's point of view? Why do things "fall down" in it despite the field gradient being zero? How can a "hill" be "flat", without being a contradiction in terms? To get the answer you need to carefully follow the differential geometry that Einstein used.

[Farsight]

Yes, I do think we're misunderstanding each other, Qfwfq. I think it revolves around interpretation. I think Einstein was right, and is right, but the interpretation of what he said has shifted. Please read pmb's paper to appreciate what I mean. Can we take it one point at a time to avoid misunderstanding? Can we start with uniform gravitational field?

[Qfwfq]

Apparently I can only see the astract of those arXive papers, but I see what he means on gravity. I honestly can't remember how much the distinction is emphasized in courses and I don't have ready access to Weinberg's textbook until I visit my old physics dep't but I really doubt it is causing researchers to commit actual mistakes. I tend to consider it a laxity in speech, the math is not being done wrong.

 

I'm the first to say that there are many pitfalls in interpreting GR and for a while I've been planning to investigate some aspects of the Schwarzschild solution. However I'm not sure this guy is pointing out something of which the best GR experts are unaware.

 

As for your friend's opinion on terminology for mass, I'm totally of the opposite camp. The choice he defends makes the word mass itself totally superfluous when instead it's handy to have a brief term to indicate a body's rest energy, a property of the body rather than a dynamic variable. To clarify my stance, I say that a composite body's mass includes the kinetic energy of its component particles (the mass of the same piece of iron is slightly greater when it is hot than after it cools down) but I don't say that each of those particles had a greater mass; they just had more kinetic energy.

 

One of Einstein's blunders was defining "direct and transverse mass" in Zur Elektrodynamic bewegter Körper.

[Farsight]

You have to click on the download where it says PDF over on the right. Try this:

 

http://xxx.lanl.gov/ftp/physics/papers/0204/0204044.pdf

 

Note that I have my own view on mass. I'll tell you about it sometime.

 

Can you give me a link to something you'd recommend on Weinberg and GR? I was reading something interesting the other day about "field interpretation" and the Schwarzschild event horizon, and need to check it out.

 

Edit: this is the "original" paper I read regarding GR. It is of course a translation into English, so there's scope already for interpretational issues:

 

http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf

[Qfwfq]

Sorry I hadn't seen the link way up over there. I always find that site somewhat disorienting. Anyway I still think this guy's efforts at teaching physicists are superfluous.

 

Can you give me a link to something you'd recommend on Weinberg and GR?

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity

Steven Weinberg

ISBN: 978-0-471-92567-5

Wiley, 1972

 

this is the "original" paper I read regarding GR. It is of course a translation into English, so there's scope already for interpretational issues:

Yeah, that's a translation of Die Grundlagen. I had a further look last night and, sure enough, he doesn't mention the word curvature, at least when discussing the Riemann tensor, but he is talking about non-euclidean geometry, using the differential tools developped by Gauss, Rieman, Christoffel, Ricci and Levi-Civita and this means that he is talking about a space-time continuum which isn't flat. What the Riemann tensor describes in this formalism is called intrinsic curvature, whether or not Einstein used the term in that paper. In it he is giving a very summary outline of differential geometry because it couldn't be considered known to physicists and he wanted to spare the necessity of going through the actual mathematical literature. You can read this at the very beginning of the paper.

 

I really believe today's GR experts don't misunderstand the distinction between the field and its gradient.

[Farsight]

Thanks for that link, Qfwfw. I'll follow it up.

 

Re the curvature, I really do think that's there is an interpretational problem that has been somewhat damaging to General Relativity. As for whether it's a problem between the experts or instead something downstream, I can't be sure. I've had conversations where Taylor and Wheeler (and Thorne) have been mentioned, but I'm not confident enough to make assertions. I think the $64,000 dollar question to ask yourself is "What is curved spacetime?" If you can explain it to your grandmother you understand it just fine. If you can't, it bears further examination. As yet I haven't met anybody who can explain to me why this aspect of pmb's paper is wrong:

 

It is widely assumed that, according to Einstein’s general theory of relativity, gravitation is a curvature in space-time. There is a well accepted definition of space-time curvature. As stated by Thorne:

 

"space-time curvature and tidal gravity are the same thing expressed in different languages, the former in the language of relativity, the later in the language of Newtonian gravity".

 

However one of the main tenets of general relativity is the Principle of Equivalence: A uniform gravitational field is equivalent to a uniformly accelerating frame of reference. This implies that one can create a uniform gravitational field simply by changing one’s frame of reference from an inertial frame of reference to an accelerating frame, which is rather difficult idea to accept. A uniform gravitational field has, by definition, no tidal forces and thus no space-time curvature. Thus according to the interpretation of gravity as a curvature in spacetime a uniform gravitational field becomes a contradiction in terms (i.e. no tidal forces where there are tidal forces). This apparent contradiction is obviously quite confusing and can certainly be misleading. In A brief history of relativity published in the above mentioned issue of Time, Stephen Hawking writes:

 

"I still get two or three letters a week telling me Einstein was wrong. Nevertheless, the theory of relativity is now completely accepted by the scientific community, and its predictions have been verified in countless applications... His idea was that mass and energy would warp spacetime in some manner... Objects like apples or planets would try to move on straight lines through space-time, but their paths would appear bent by a gravitational field because space-time is curved".

 

Given such a statement by a respected physicists and with a great deal of experimental data to back up this claim is it reasonable to question this notion of gravity being a curvature in space-time? Is it reasonable to assume that the bent path an object takes when moving through a gravitational field is due to the space-time curvature? Did Einstein actually hold the view that gravity is a curvature in space-time? At this point let us whet your appetite. No. Einstein never said nor implied in anyway that gravity is a curvature in space-time. This apparent disparity is a result of a change of interpretation. However it would be a great injustice to imply that there is no relationship between gravity and spacetime curvature. Curvature plays a very important role in general relativity and its importance should not be underestimated. However the aforementioned change in interpretation is most likely the source of various errors in the scientific literature...