I'm trying to find a way to find the amount of elapsed time taken place upon a body going through acceleration. If we are given uniform acceleration, say v=t+C, then I think we can rewrite this as

There is always a problem with the question: 'can SR handle acceleration?'. The answer is yes and no. SR is perfectly capable of dealing with accelerating objects that are seen by observers in inertial frames. Indeed, if you differentiate the Lorentz transformations twice, that is exactly what you will get. (I don't think that is what you are after though).

On the other hand, SR says nothing about what accelerated observers might see. This is the sense in which the General theory is required to deal with acceleration. Note that there is only one parameter in the Lorentz transformations: the relative velocity of the two inertial frames considered. Were SR capable of dealing with accelerated observers, there would have to be some acceleration dependence in the transformations themselves, i.e. another parameter.

In physical terms, what I think you are doing is trying to approximate to an accelerated observer, by treating him/her as a succession of inertial observers at different velocities. You then take the limit of 'time spent at any one velocity' -> 0. This seems like a sensible thing to do, but there is really no reason why it should work. Remember that at each stage, your approximation is already unphysical, because you are assuming that the step from one velocity to the next is instantaneous. So, I doubt that the result from integration will correspond to experiments with real accelerated observers.

I don't think there is any way to solve the problem in the context of SR. Say you made your approximation more physical by having sudden, but finite accelerations between the velocity steps. You would have solved one problem, but got nowhere, because you would then need to know exactly what you are trying to find out, i.e. the time-dilation formula for accelerated observers.

As it goes, SR can sort of deal with physics from within accelerated frames, the way it is done is to consider the accelerated frame as a series of instantaneous non-accelerated frames, but it is somewhat artificial and GR is needed for a nice solution.

Brad - I don't really understand your notation, but one point is that the acceleration v=t+C is not a valid acceleration because the body will end up going faster than light.

The formula you want is, I think, that the proper time elapsed is (t here is proper time)

V^m will be given by the force law.
Sean

I'm not sure if calculations based on the approximation of a series of inertial frames actually produce the correct answers.

Simon - It does work and is in no way an approximation, but it only gives you instantaneous information. So for example, it couldn't give you the trajectories of objects (which require an integration over time), but it can give you, say, what an accelerated observer sees as the instantaneous velocities of other objects.

Sean

I hadn't thought of the fact that my object will exceed c, but for the moment let's just assume that t+C < c. Is there a way to integrate my original equation(or indeed Sean's, though I don't understand it as well-I'm just beginning to understand four-vectors and the like)? Or is there even a point.

Why wouldn't acceleration as a series of seperate velocities work? I am not sure I entirely understand why it wouldn't, so long as the acceleration is uniform. I am not trying to find out what the two observers see, just a proportion of the times they go through.

The reason I wanted to find this out was to find a resolution to the twin paradox, and if that integration integrates to something even near what I think it would, I don't see much of a resolution to the paradox, so maybe you are right, but I still don't see why. I'll perhaps have to learn GR first though.

Also, the reason I was trying to figure the amount of time for the accelerated observer is mainly for simplicity. This perhaps, was my error. But, could we still intgrate that to form an idea of the amount of time the inertial observer sees the body go through(not the actual time traveled through)?

One last question: anyone know of a good book to learn tensors from? I purchased D'Inverno's book, and the SR section was written extremely understandably (though I'm still working on mastering some of the topics). But then I got to tensors. The fastest I can move is a page a day. So, is there a better book to learn from, or will I just have to be content with finishing tensors in January?

Thanks,

Brad

Actually, D'Inverno's book has one of the easier treatments of tensors. Don't worry if you find this difficult, it is conceptually quite different from things you have probably done before.

I think you will find it easier if you read the SR section on Four-vectors thoroughly, you will not be able to move onto GR until you understand four-vectors.

The problem with the treating acceleration instantaneously is that you have to use different Lorentz transforms at each time instant to transform into the accelerated frame. But it does work for a given instant. It is not useful though for doing the integration you want to do.

The twin paradox CAN be resolved without GR. The best way to do this is to draw some spacetime diagrams, I think this is done in D'Inverno, although I may be wrong.

Sean

Yes D'Inverno's book does resolve the paradox through spacetime diagrams, but the argument he uses doesn't quite make sense to me. It seems as though the accelerated observer could draw the same spacetime picture relative to himself and thus the paradox is as it was before. Is the reasoning behind the resolution that in SR acceleration is absolute and not relative.

I don't know that it really matters, but using clock hypothesis, I was able to integrate my equation above to

T=(1-(t/2+v₀ /c)² )^1/2

Where T is once again the proper time. Then, is the clock hypothesis wrong?

Brad

The clock hypothesis is not wrong.

The spacetime picture for the accelerated observer is not the same. When there is an acceleration the observer is no longer inertial so you have to change frames. So if you want to do the calculation for accelreated observer you need to do it in two bits. This is why most discussion of the twin paradox is done with a single instantaneous acceleration, a more continuos treatment is not easy in SR.

If you do the calculation in two bits like this, you find exact agreement with what you would find using the other frame.

Perhaps you could explain where you got your original expression from, I suspect it involved an invalid use of accelerating frames.

Sean

I don't think a continuous treatment is possible in SR. Spacetime diagrams seem quite dubious too. There's always this curvy bit where the acceleration is supposed to be taking place. The whole point of a space time diagram is that you can just transform the axis, and then look at the whole motion (i.e. the same worldlines) from another observer's point of view. This is how we get the wonderful graphical explanation of how two observers in relative motion can both see the other foreshortened. However, this only works when the worldlines of the observers are straight. The curvy part seems like an ad-hoc addition that serves no purpose other than to solve the twin paradox.

I'm not sure that the argument of an instantaneous velocity change is at all valid either. If we won't allow acceleration, then infinitely quick acceleration is not much better.

The only way I can think of to solve the twin paradox within the confines of SR is by setting up an exactly equivalent problem that doesn't involve acceleration:
First, twin 1 is passed by twin 2 moving with relative speed v. Twin 2 goes on its way, and then meets twin 3, who is travelling in the opposite direction, back towards twin 1. But, when twins 2 and 3 meet, they synchronise their clocks. So, twin 3 meets up with twin 1, and they compare the readings on their clocks.
Here there is definately no problem in treating the motion in 2 parts, since it is 2 separate objects that are involved.

Here was my logic used:

There is no doubt that the increase in proper time is equal to the inverse of the rate at which moving clocks slow multiplied by the increase in inertial time.

Thus dT/dt=1/b
Because the velocity changes at a rate of t, we arrive my above equation apart from the flaw that v₀ +dt should be v₀ +t. Now the clock hypothesis relies upon the fact that the average velocity used in the calculations is the same as taking into consideration all velocities over infinitesimal increments of time . With this, it holds that my original equation integrates to

T=(1-(t/2+v₀ /c)² )^1/2 t

(I forgot the ending t before)

I have not checked the differential of this, so I'll do so right now, thus disproving or proving my conjecture on what the clock hypothesis is based on(for v=t+C). It looks from where I am currently as though the two are incompatible, though I don't see why. I would have thought that they were saying the same thing as each infinitesimal velocity multiplied by the amount of time that it occured at from zero to t would equal the same distance as the average velocity times t. But perhaps that's where I'm making my mistake. Or maybe I'm making no mistake at all (ha!, I like to amuse myself every once in a while). As it is, it's late (hence how incomprehensibly I'm writing) and I have to wake up early, so I'll write in with my finished calculation tomorrow.

Brad

Deciding to just go ahead and do the raw algebra, I have found that the two methods of calculation are indeed not equivalent! From this solely, we can not conclude that even either of the methods are true, just that one must be wrong. But, taking into mind expiriments, one must be forced to conclude that my method stated originally, is not true. But why then, is my method wrong? If given info for instantaeneous rates, why can't we figure total rates? It just doesn't seem to add up. I'll check my results in the morning.

Brad

Brad, I think there are 2 reasons why your method didn't work. One is that as Sean says, v=t+C doesn't make sense from a relativistic point of view. We know that no observer can measure the speed of any object to be > c, and clearly t+C will be > c for sufficiently large t.

The other problem is exactly the one you highlight, i.e. that although you have information at any particular velocity, you can't integrate this in the normal way. This is because of the definition of the integral as the limit of a sum. Here's an example of another problem that can't be solved in this way. Consider a rocket accelerating. Say we want to know how much fuel is burnt. We know that fuel is only burnt when the rocket is accelerating. So, at constant velocity, no fuel is burnt.

So, apparently,

Simon - The last method you mentioned is obviously the best and works. However, you really can do it with two observers only in SR using an instantaneous acceleration. Physically dodgy this may be, but I guarantee the spacetime diagrams give you the right answer in whatever frame you do it. I would write it down here but it's a bit long and requires the diagrams.

The way you would have to do continuous acceleration is to use different Lorentz transformations at each instant, doing this basically gives you GR! (or at least the basics of it - see Misner, Thorne and Wheeler, the big black book, the chapter on acceleration in SR)

Sean

It kind of depends on who you ask. There is a sort of 'if it gets the right answer then that's all we need to know' trend in physics at the moment. So the physicists I know have no problem with using space-time diagrams in this way. On the other hand, the lecturer of the philosophy of physics course had other ideas! But then they thought that all sorts of tiny details needed arguing for, e.g. the fact observers travelling uniformly in opposite directions measure the same speed for each other (principle of reciprocity).
Thanks for the reference. Is the big black book good? The one I have been using for GR is the smaller blue book by Weinberg.

Simon

I think the 'trend' is hardly new, but has been around as long as physics has. It tends to be the right sort of attitude that allows you to break new ground and make discoveries, because when dealing with something new, it is inevitable that not everythign will be consistent right from the start.

However, to SR, it is interesting that I recently flipped through a book written as late as 1970, called The Logic of SR, the basically contained a debate between eminent physicists as to whether the the twin paradox was resolved and mnay seemed to think not!

But even so, I don't think there is anything unrigurous or unphilosophical about using spacetime diagrams, they are just a graphical representation of the mathematical situation which are useful because they provide a visualisation.

The big black book is good as reference, I don't like the way it does tensors, and it is far to long to try and learn from. I would have thought the Weinberg book a little long to learn from also, but maybe not.

Sean

There was a Prof. Dingle of somewhere or other who apparantly maintained that:
1. the moving twin would be younger on arrival
2. SR doesn't predict this
3. so SR is wrong
I can't remember his justification, but apparantly he was quite eminent; although his reputation suffered a downturn.

Space-time diagrams are perfectly good graphical representations of many situations. But they only have the useful properties they do under certain conditions.
The axes are distance and time, and so are particular to some reference frame or another. The advantage is that if you want to look at the situation from another inertial observer's point of view, then you don't have to alter the diagram, you can just refer to it from a set of axes oblique to the first one. But this only works if the observer you want to switch to has a straight world line, i.e. is inertial. Where the line curves, you would effectively have to use different axes for each point, which needless to say would remove the elegance of the diagram. This is exactly analogous to using a different Lorentz transformation for each instant.

I think the problem with the spacetime diagram explanation of the twin paradox is that when you draw the lines representing the signals from one twin to the other, you don't really know how densely to put the signals from the moving twin to the inertial one during the period of acceleration. You can't just place them evenly along the line, because you don't know how the curve transforms to the straight line that it would be when viewed from the point of view of the accelerating observer. What you find in 'explanations' in books is I think just a fudge, i.e. they put in as many or as few signals as they need to to solve the twin paradox.

Yes, Prof. Dingle comes up a lot in the book I mentioned.

The only problems with the spacetime diagrams is that to use them properly in this case you need a point acceleration. This is unphysical, agreed, but if you look at it as a limiting process of more and more instantaneous accelerations (something like you normally treat a delta-function in physics), I think the limiting process is valid. So the curved part of the world line becomes smaller and smaller and hence less important.

Basically, I'm trying to say that the instantaneous acceleration isn't a fudge, but a convenient limiting process that we use in all sorts of situations in physics, pretty much whenever delta functions come up.

A clear way of looking at the situation is to note that the proper time elapsed for an observer is proportional to the length of the world line (this is the equation I wrote a few posts ago), however, we are in Minkowski space so length has a minus sign when it is in one of the directions. So in fact the longer line on the spacetime diagram is the line for which less time has elapsed. This is why the inertial observer sees that the accelerated observer has aged less.

Now for doing the reverse calculation (i.e. from the accelerated observer), we need two diagrams, one for before the point acceleration and one for after. In both this diagrams both observers are straight lines. However, if you actually draw them you find that the total length of the line due to the inertial observer is longer. So more time has passed for him and there is no paradox.

(some pictures would have helped that explanation, hopefully you can draw them...)

Sean

The example was only really relevant to the case of continuous acceleration. If we accept the validity of the point acceleration model then I think the space-time diagram explanation is fine. I don't think that it is acceptable though. You say the curved part becomes smaller and therefore less important. Why should this be so? After all, as the time spent accelerating becomes smaller, the degree of acceleration has to increase for the moving twin to get back in the same amount of time. The comparison with delta functions presents another problem. Say we define it in the usual way (slightly simplified):

I am pretty sure the last sentence is wrong. It IS true ALWAYS that the length of the world line, understood as

... it is in fact a delta function, not just a comparison.

How do you know it is always true that the length of the world line is the proper time? If this is true for all worldlines, then surely that's all we need. It's easy to show it for straight worldlines, as the time axis of the spacetime diagram is just rotated under the Lorentz transformation.

Ok, so in the point acceleration approximation, you just have a world line divided into two parts, both of which are straight lines. So you can calculate the proper time along it. But, consider one of the steps towards this limit, i.e. a small period of acceleration. How do we know that the proper time will tend towards the value given by the instant acceleration approximation?

The analogy I am thinking of is with the length of the diagonal of a unit square. We know it is sqr(2). It is also the limit of the following series:



All of these lines have length = 2, even though they tend towards the diagonal with length sqr(2).
So, if the real physical situation is some way towards the limit, and all we know is the value at the limit, then we don't really know what will happen.

The series you give doesn't converge to the expected value because of the way it's been constructed (basically to do with the difference between discrete and continuum ideas). This does not happen here because if we agree that the length of the world line gives the proper time, and I will prove this in a minute, then it is clear that the proper time associated with the acceleration is getting smaller and smaller and tends to zero, there is no problem with the limit.

Now, the proof, in some sense it is obvious from the 3D analogue that

__________ Ödx/dt dx/dt

dt = dx

(in fact, this is exactly what is happening). In another sense it is by definition, because all the proper time is is a way of parametrising the world line and in fact the integral in my previous post is invariant under reparametrisation t \tot ' (t). Here t is the parameter along the world line, it is called the proper time if it is such that dX^m/dt gives the 4-velocity V^m which satisfies V^mV_m = c². Which leads us to the proof, because it implies the integral becomes ò(c dt) = c t_tot = c × total proper time.

Sean

Ok, I agree your proof works. So why bother at all with the point acceleration business?

Also, in a sense, your integral is giving something additional to Einstein's postulates: It is ruling out any extra explicitly acceleration-dependent phenomena. So in that sense, it is extraneous to SR, although to be honest, I'm getting less and less sure where the borderline lies between SR and GR.

The point acceleration limit is useful because in this case we can use spacetime diagrams, which we can't do while the particle is accelerating over a finite time (well, we can, but we can't do them from the point of view of the accelerated observer, which is what we want to do to reolve the paradox).

In a certain sense it is true that acceleration is not mentioned explicitly in Einstein's postulates. However, once you have set up Minkowski space, you can construct 4-vectors, including 4-acceleration, and this doesn't involve any new physics. But perhaps you are right that there is an additional postulate which is that an accelerated observer is instantanously an inertial observer... I'll think about it.

The boundary between SR and GR is reasonably clear, I think. In SR the metric is the minkowski metric and in GR it is a more general metric, that is determined by the GR field equations and which influences motion through the geodesic equation. Everything else is the same.

Sean

whooa, sorry, I forgot to add that in GR you have the equivalence principle which tells you that you can transform into the accelerating frame and gives you the transformation to do so..