Here’s an alternate version at project Gutenberg, this one has pdf-versions including the illustrations: Relativity: The Special & the General Theory by Albert Einstein - Free Ebook
So I did just read some of the book on project Gutenberg (thanks Naita). I find it amusing that he starts the relevant chapter with:
We imagine a large portion of empty space, so far removed from stars
and other appreciable masses…
which is flatly in contradiction with tomh4040’s examples involving black holes. But nonetheless, as I showed in my previous post, even in his example things work out just fine.
But stepping back, I’m curious what tomh4040’s real problem is here. I haven’t read the entire thread, I’ve sort of jumped in, but the focus on a corruption and misapplication of one of Einstein’s examples seems like a distraction. Look, the equivalence principle is actually pretty simple. It says that if you put an object in a locally flat gravitational field, all of the atoms in the object will be accelerated at exactly the same rate. It just so happens that this means that you won’t notice anything if you are in free fall. This means that you can be at rest inside a chest, or in a chest accelerating towards a planet, and everything will look the same. This means that exerting a force on a chest in a gravitational field will be experienced the same from the inside as exerting a force on a chest far away from a gravitational field. In one case, the force may keep the chest from accelerating in a gravitational field (the chest is on the surface of the planet), and in another case the force may accelerate the chest in the vast expanse of space. Everything simply follows from the fact that all of the atoms in an object will be accelerated at exactly the same rate in a gravitational field, regardless of their mass. This fact has been experimentally tested to thirteen decimal places. What’s the problem?
tomh4040 thinks RT is wrong, or at least flawed. He’s basing this on reading someone’s flawed arguments against it. He started out with a convoluted attempt at proving the twin paradox doesn’t exist in reality.
I read through the thread and came away with the impression that we need to discuss a sense in which popular descriptions of relativity suffer from promising more philosophical satisfaction than they can deliver. I myself have been disturbed by this before; both because of the feeling of loss at realizing it is not as beautiful a theory as I had first thought, but also because most other physicists don’t seem the least bit interested or concerned by this loss. It may be because they were never seduced by the same misimpressions I had, or it may be because they are so doggedly pragmatic and (sometimes tiresomely) reactionary to philosophical inquiries (we are scientists not philosophers!), that for them it is simply not an issue worth considering. Anyways, I’ll air my two anxieties, because it might help both the OP and tomh4040 get over their own apprehensions.
The first concerns SR and the twin paradox. The misimpression I had was that the idea of relativity ran much deeper than it actually does. Philosophically I was seduced by the idea that really, in a deeper sense than is true, the universe itself cannot distinguish between two inertial reference frames. That it cannot “keep track” of objects in any global sense because there is no global coordinate system in which the universe can keep track of their trajectories (ie, which twin travelled and which stayed put). While it may be that one of the twins experienced acceleration some of the time (the length of the acceleration and its effect on time dilation can be made negligible), this does not “break the symmetry” if the rest of the time the universe cannot “tell” because it isn’t keeping track (how would it if there is no “real” background reality it can “plot” trajectories on?). When we resolve the twin paradox by drawing the world line of each twin and showing that the twin that accelerates has a different arc-length, we are doing something that the universe can’t (so the thinking goes); we are plotting trajectories on some fixed background, using some preferred frame. We could have done the same calculation using another frame and gotten the same answer, so while it is true that there isn’t a preferred frame, there is still a preferred class of frames that informs the universe’s bookkeeping. So which is it then? What is the fundamental reality? Is there some background on which objects are “tracked”, be it the “class of inertial frames” (the Poincare group), or are things really relative? The ultimate problem is that our modern understanding of SR eschew’s the beautiful Machian insights that originally motivated Einstein in the first place. When I first learned about relativity, I had the impression that the principle was so ultimate, that, for instance, if the earth is rotating relative to nothing in an otherwise empty universe, then it really isn’t rotating at all, since there is nothing relative to which it is rotating. But the solution to the twin paradox tells us that the answer is yes: the earth can rotate even if it is relative to nothing. In other words, it’s not all relative. There is some background against which the universe keeps track of things. And this is not as philosophically satisfying as I had first thought relativity would be; there is a preferred something, it just doesn’t happen to be a single inertial frame. So instead of harping on the issue of relative motion, which stirs all sorts of philosophical biases, I think it might be most pedagogical to think of SR as merely the consequences of there being a finite speed of light and the laws of physics being covariant on top of a real Riemannian geometry.
The second concerns GR and the equivalence principle. When I first learned about the equivalence principle, I thought it really meant equivalent. But it doesn’t. It doesn’t mean that if you are held fixed in a gravitational field it is really in all senses equivalent to if you were being inertially accelerated, despite how seductive of an idea that might be. All it means is that gravitational and inertial mass are equivalent. As others have pointed out, the difference in time dilation is one example of the inequivalence. The point is that there really is a reality to the curvature of space that cannot be transformed away by appeal to a philosophical notion of relativity. And this ties in with the first concern: that there really is some metric against which things are judged in a non-relative way, and properties do not bootstrap into existence through relative motion.
At the end of the day, just like in quantum mechanics, we have a theory, it is unintuitive, and it works (extraordinarily well). It is unfortunate that much of the popular discussion of relativity has not completely abandoned its Machian heritage (Mach was wrong).
I probably didn’t make myself clear, so I will make this point again.
Einstein’s statement in chapter XX is correct. The mass, when released, is simply left behind as the chest accelerates away, so will fall at 9.8m/s^2 .
Einstein’s statement in chapter XIX is wrong. The mass falls at the combined acceleration of the two masses, which in the case of the black hole is 19.6m/s^2 . In the case of the wood or lead, the acceleration is so close to 9.8m/s^2 as to be indistinguishable from that figure.
This explains why the piece of wood and lead appear to fall in exactly the same manner. Their mass is so tiny compared to the mass of the Earth, that any difference in acceleration between the two simply cannot be measured. If a black hole falls faster than a piece of lead or wood, then so does a mass half of that of the black hole, or a quarter, or a tenth etc. In principle, if the man’s instruments are sensitive enough, he can tell whether he is in a gravitational field or being accelerated (by a hypothetical being).
Note the sentence “If a black hole falls faster than a piece of lead or wood, then so does a mass half of that of the black hole, or a quarter, or a tenth etc”. I was using the miniature black hole as an example, and ignoring tidal effects, which I do know about, even though some of you think that I don’t, in order to lead us on to the fact that objects fall in proportion to the combination of the mass of the Earth and the mass of the falling object. It is easy now to see that he lead actually falls faster. This led to one posting saying that there are therefore two gravitational fields; one for the lead, and one for the wood. That is perfectly true, and gives us an insight into Einstein’s muddled thinking. He says that the lead and wood fall in the same manner in A gravitational field (my capitalisation). They are bound to fall at different rates (though the difference is minute and cannot easily be measured) simply because of that fact. When in an accelerated FR however, they fall at exectly the same speed - they are simply left behind as the chest accelerates away.
For those of you who want figures, a 1Kg weight suspended 20 meters above the Earth’s surface will fall in :-
2.0196199771025520756348344841573 seconds.
A 1000Kg weight will fall 20 meters in :-
2.0196199771025520756346658201616 seconds.
Both examples neglecting air resistance of course.
This is a difference of 1.68663995e-22 seconds. Not a lot, but it is there.
There is nothing muddled about Einstein’s thinking. He is, for the sake of clarity to a non-technical reader, examining what is called a limiting case. In physics, this is extraordinarily common even in technical settings. It is, in fact, difficult to find cases where these kinds of simplifications are not made. Some assumptions, like ignoring the gravitational field of small test objects, is generally taken for granted. Show me any equation from statistical mechanics, classical mechanics, quantum mechanics… and I will show you where some approximation is being made. Newton’s laws? Approximate. Ideal gas law? Approximate. Standard Model lagrangian? Approximate. This is not due to muddled thinking; those who are literate in the field immediately recognize the implicit simplifying assumptions that give the equations a range of validity. The equivalence principle is a very normal example of this: the simplifying assumption is that space-time is locally flat. This means that it has a range of validity, applying to small, non-dense objects. Einstein was well aware of this. This is why, in his examples, he doesn’t use black holes. When you do, you have to include the object’s gravitational field in “A gravitational field” when doing calculations (Einstein’s wording is perfect here, because there really is only A gravitational field even when there are multiple objects contributing to it). In this case this is no big deal at all; as I showed earlier the theory is perfectly consistent as long as you remember that, if the test mass being small is no longer a valid approximation, then you should go ahead and include it in “A gravitational field”.
Quote from the book.
The present book is intended, as far as possible, to give an exact
insight into the theory of Relativity to those readers who, from a
general scientific and philosophical point of view, are interested in
the theory, but who are not conversant with the mathematical apparatus
of theoretical physics.
Note “exact insight” and “not conversant with the mathematical apparatus”.
This does not mean he is talking to dummies, as one reply inferred. I do have to admit that I fall into the category of “not conversant with the mathematical apparatus” in some respects. I get there in the end, but slowly; and I prefer the written word to maths. Don’t forget that originally, all calculations were done in the language of the writer, and could be quite lengthy. Mathematics was and is a device used to shorten and unify calculations done in different languages. Now here we come to an interesting point. Mathematics became a language of its own, and instead of being subordinated to physics, physics became subordinated to mathematics. It came to be thought - erroneously, that if something was mathematically correct, it was physically correct. That is not so.
An obvious example is the use of dimensions. In maths (and in computing), there can be as many dimensions as wanted. In the real world, a specific location in space and time can be found by using four, and that is all there are.
Einstein’s SRT may well be a mathetically correct model, but it is not physically correct.
And as for the posting that states “I read it somewhere”, guilty as charged. We all read it somewhere, as none of us were involved in the formulation process of anything under discussion here. Here is something which I read, which changed my mind about SRT. At one time I was a believer, but not now.
Professor Dingle, once highly regarded in relativity circles, so much so that he was asked to write, and did write a book on the subject, and gave Einstein’s eulogy,wanted one question answering. This is it :-
“…you have two exactly similar clocks…one is moving…they must work at different rates…But the [SR] theory also requires that you cannot distinguish which clock… moves. The question therefore arises… which clock works the more slowly?”
This is in the introduction to his 1972 book “Science At The Crossroads” where the question is in full. I have lost the full version. The web page this was taken from is :- http://quantumfieldtheory.org/Dingle.pdf
So now your “argument” has devolved into questioning Einsteins semantics in expressing how much detail he intended when writing for a lay audience. Nice. I guess you’ve done it. You’ve disproved special relativity.
I thought this was a civilised discussion, not a sarcastic put-down.
The wheel has turned full circle, this is where I came in. I can’t make out where you stand on this matter, as reading your postings, this is not made clear.
Please answer the following :-
Which of the two clocks is moving? Why that clock and not the other?
Is a fast moving IFR really experiencing time dilation, is more massive, and is contracted in the line of motion? If so, is a stick half in water really bent or not?
If a rocket carries its own internal propelling force ie a reaction rocket motor, can it accelerate past c?
It depends on the frame of reference.
If we’re moving relative to each other:
[ul]
[li]From my perspective your clock is running slow.[/li][li]From your perspective my clock is running slow.[/li][/ul]
If we ever wind up back in the same reference frame, which clock is ahead and which clock is behind depends on the specific accelerations we used to synch our frames.
From its own reference frame, no.
From the perspective of an external frame, yes.
No.
At a certain point an expert in the field (at least a near-expert with regard to SR/GR specifically), who enters into an argument in good faith, trying to dispel obvious confusion, becomes so exasperated at responses that don’t AT ALL ADDRESS ANY OF THE POINTS HE HAS SPENT THE TIME TO MAKE, that he gives up. It has happened already to others here trying to help you. Now it has very nearly happened to me. It might be a signal that there is something wrong with how you are engaging in conversation. Look back at my posts, and you will see that I have very clearly answered some of the questions you have asked, and you have responded directly to NONE.
The clock that accelerated is “moving.”
Relative to what? There is no such thing as the “experience” of time dilation, nor of the “experience” of length contraction. From your “experience” it is other things that look time dilated and contracted. But the universe keeps track of who is moving more, so at the end of a trip you might see other things as older or younger if you meet up again.
Nope.
You cannot just answer nope. The propelling force is in the rocket, the relative velocity between rocket and force is therefore zero. Do the Lorentz equation for mass increas (et al) with v = 0, and there is no mass increase. The argument most often heard is that nothing can travel faster than light because mass increases without limit with increasing velocity, so the propelling force has to be increased without limit also. That argument does not hold.
For a rocket with constant proper acceleration α, the coordinate acceleration a, as observed by an inertial (non-accelerating) observer is given by
a = α/γ[sup]3[/sup]
where γ = (1-v[sup]2[/sup])[sup]-0.5[/sup]
where v is the relative speed of the object with units choosen so c=1.
From that equation it’s easy to see that a->0 as v->c whatever the value of α is.
We needn’t relate this to the moribund concept of mass increase (as has been said earlier the definition of mass which means that it increases with speed is no longer used by physicists), it’s simply to do with acceleration transforms under a Lorentz boost.
How do you have a velocity relative to a FORCE? What does that even mean? :dubious:
Here’s what happens from the perspective of the rocket:
As it goes faster and faster the rest of the universe becomes more and more Lorentz-contracted. Eventually (as the rocket approaches light speed from the perspective of an external observer) the entire universe will seem to be compressed to an infinitesimally-thin disk. The rocket can’t go any faster, not because it’s too massive for the engines to move, but because it’s already gone everywhere it can go.
Relativity describes how observations differ according to your reference frame. Since both rocket and engine are in the same reference frame, applying relativity to the relationship between them is nonsense.
As The Hamster King explained, what you’d expect is a change in observations of the universe from the rocket and of the rocket from the rest of the universe.
What relativity tells us is that as observed from an inertial reference frame the rocket’s kinetic energy will asymptotically approach infinity as it approaches the speed of light relative to the inertial reference frame. The observations from the rocket will be that it’s the inertial reference frame that changes.
You still have very basic misconceptions about relativity. Have you tried reading some modern introductions instead of Einstein’s text? We’ve learned something about what misconceptions are most likely to occur in the 100 years since relativity was introduced. Why not benefit from that? Relativity: Einstein's theory of relativity in animations and film clips. Einstein Light
tomh4040, your latest post seems to be answered in spirit by my post #245. Please read it. The fact is that the universe does keep track of motion, despite the popular misconception that the equivalence of inertial reference frames implies that there is no way for the universe to possibly view the trajectories of the two twins differently. Motion is relative to a Lorentzian Manifold. The motion is all kept track of (ie which twin is moving more than which other twin). I just happens to be a property of the spacetime metric that there is a symmetry between the class of inertial reference frames. But there is no such corresponding symmetry between all trajectories over the manifold. That symmetry is broken by acceleration. Special relativity allows for and accounts for acceleration, despite your misconception. See, for instance, the equation for proper time here. But even if it didn’t, the fact is that SR is not the complete picture, so relying on it totally in trying to solve your “paradox” is imprudent. The correct theory is called General Relativity.
I have read it, more than once in fact, and that is what prompted the question. You appear to be more of a philosopher than a physicist, in that your posting contained a lot of words but said very little, which is common amongst philosophers. You argue for relativity, yet use a preferred FR when the need arises.
To answer my own question. A rocket can exceed c, and here is one reason why. This was originally said by H Tilton, but is not a direct quote.
Photons cannot exceed the speed of light. C is the fastest that anything can travel, including information. Photons also cannot travel slower than c. The rules which govern solid bodies are demonstrably not the same as the rules which govern photons, as solid bodies can and do travel slower than c. So as they are not constrained to a lower limit, why should they be constrained to an upper limit?
A rocket ship does not present the same velocity profile to all observers as photons do - a crucial distinction that makes all the difference. The speed of a rocket ship depends on who measures it: the speed of a photon does not.
But time and distance also depend on who measures them.
Say I’m riding a rocket to Alpha Centauri and I’m traveling at 87% of the speed of light relative to Earth.
From the perspective of someone on Earth, Alpha Centauri is 4.3 light years away. And from their perspective my clock on the rocket is ticking slower. At 0.87c it’s ticking about 1/2 the speed of a clock on Earth.
Now, from my perspective on the rocket, my clock is ticking normally. However, the rest of the universe is Lorentz-contracted, so the distance from Earth to Alpha Centauri is cut in half. They’re only 2.15 light years apart.
Everyone can agree on how many clock ticks it takes for me to get to Alpha Centauri. However some observers think it’s because of time dilation and others think it’s because of Lorentz contraction.
I can’t travel faster than light because when I reach c my clock stops ticking entirely (from the perspective of Earth) or the universe contracts into a 2-dimensional disk (from the perspective of the rocket). Either way “traveling forward” becomes undefined.
Well, YMMV. To be clear though, I am a physicist, not a philosopher. My post was specifically meant to address what seems obvious to me (whether you realize it or not) are your philosophical misgivings. For instance the post (as well as the most recent one you replied to) both specifically address your repeated pleas about “which twin is really moving.” Yet you do not reply to those points. Your statement “You argue for relativity, yet use a preferred FR when the need arises” makes no sense to me, as I do not such thing. It is clear that you still possess a profound misunderstanding (as is also demonstrated by your incoherent argument about exceeding the speed of light) of SR and should really start from the beginning of a standard textbook before continuing to ask your questions here.
But all changes in perceived velocity are bounded by the Lorentz equations, which never produce a number for v that is greater than c.
The key term to the equations is the square root of (1 minus (v-squared over c-squared).)
(I apologize for not knowing how to display algebraic terms… It’s much prettier when seen in the full notation. Here is a link that shows the equation.)
The key point is that there is nothing you can put in to the equation that will allow anything greater than the speed of light to come out of it. No one ever sees a rocket going faster than light, no matter how fast they (the observers) might be going, in whatever direction.