Markus Hanke

I'm in agreement with you, elfmotat. That is pretty much what I was saying anyway; it boils down to the fact that there is a difference in metric tensors between the observer at infinity and the observer inside the cavity, even though the actual values are not fixed and simply are a result of the boundary conditions imposed. Hence the time dilation. So I think we are all good

That's true, but you need to consider the boundary conditions of this problem as well. The metric inside the cavity needs to connect smoothly to the interior metric of the shell itself to ensure global differentiability, which then again needs to connect smoothly to the exterior Schwarzschild metric. It would be interesting to do the numbers here, but just by looking at it I don't think it will work if the metric in the cavity is Minkowskian. If that is the case we will get either a discontinuity at the boundary, or a hypersurface where the metric is smooth but not differentiable, both of which is unacceptable. I don't believe it is possible for the elements of the metric tensor inside the shell itself to be exactly +/-1 at any point including the boundary, since this would imply a Minkowskian vacuum with vanishing energy-momentum tensor; so, in order to maintain smoothness and differentiability at the boundary, the vacuum metric in the interior of the cavity cannot be +/-1 anywhere either, or else we have a boundary problem. But then again, that's just my two cents' worth from the point of view of differentiable manifolds, I might well be wrong. Has anyone got any references to a fully worked calculation for just such a case ? I couldn't find anything. I think here's the solution : locally the metric inside the cavity is Minkowskian; however, if we want a global coordinate system which asymptotically approaches Minkowski at infinity, then the cavity will be flat, but not Minkowskian. This takes care of the time dilation issue, and tallies nicely with Newton.

Having thought about it further, it would appear I made a serious mistake here, so I have to retract the above statement. We will indeed find a region of completely flat space-time, but not the Minkowski metric; instead, we will have a metric tensor the elements of which are all constants, but not equal to +/- 1. These constants will be some function of the shell mass and the cavity radius. Physically this means there is no gravitational field in the cavity, but a clock located there is still dilated compared to a clock located at rest infinitely far away. This tallies nicely with the Newtonian shell theorem, and in terms of classical mechanics, can be thought of as a non-vanishing gravitational potential inside the cavity. It should be noted that there are no potentials in GR, it is the metric tensor which is the source term for gravitational time dilation.

Take the first sentence for example : This is without any meaning.

I am afraid you are using too much non-standard terminology for me to give any meaningful reply; I quite simply have no idea what you are trying to say.

Yes, I agree, and that is currently our problem. The domain of quantum gravity is far beyond the reach of any experimental setup which we can conceive of. This may change in the future, but it's what we are up against right now.

So far as gravitational time dilation is concerned, the important term is the metric tensor. In the case of the cavity within the shell, we simply have the Minkowski tensor [math]\eta_{\mu \nu}[/math], which represents a completely flat region of space-time. Btw, the Newtonian potential of the gravitational field in the cavity is constant at all points, and equal to the potential at the surface of the shell. This represents an extremum of the potential energy function; the numerical value - and thus whether this is a minimum or a maximum - is simply a matter of convention. The important point is that a stationary observer inside the shell will, as compared to a stationary observer outside the shell, age faster, due to the fact that his region of space-time possesses no curvature. Observers outside the shell are in a region of curved space-time, and thus age more slowly due to gravitational time dilation.

Ha ha, this is brilliant !

This is an old idea, and trivially wrong. Gravity and electromagnetism do not behave in the same fashion, and cannot be described by the same laws. In fact they differ in pretty much all aspects, most notably that one of them is a vector field and the other one a tensor field, and that gravity is self-interacting whereas electromagnetism is not. While it is true that electromagnetic fields are one possible source of gravity, so too are all other forms of energy. All forms of energy are sources of the gravitational field, not just electromagnetism. Btw, the term "strong electromagnetic force" is physically meaningless. You have electromagnetism, and you have the strong interaction. They are physically distinct phenomena, and not the same thing. Again, these interactions behave in completely different ways. I should also remind you that this section is not the place to present personal theories; the moderators will probably not take kindly to that. I suggest you open your own thread on this in the appropriate section, but you will find that the idea is old and has long since been shown to be wrong.

The ability of GPS to triangulate positions is dependent on clock synchronisations. A GPS satellite in orbit and clocks on earth are subject to both relative velocity time dilation and gravitational time dilation. The latter is a result of GR, and induces a difference in proper times to the order of 35ms each day. The GPS system compensates for this effect - if it didn't, all positions determined by it would be off by roughly 10km each day, so accounting for GR effects is crucial in making GPS work. This compensation is built into the software on all GPS receivers, since the orbital parameters of the satellites are fixed and thus the amount of gravitational time dilation is always known. Space-time is modelled as a 4-dimensional pseudo-Riemannian manifold endowed with a metric and the Levi-Civita connection. How that model relates to physical reality is a largely philosophical question which I will not attempt to answer here. Suffice it to say that the predictions made by the model are in good agreement with experiment and observation, or else GR would not be part of mainstream science. Gravity is an intrinsic geometric property of aforementioned manifold; in classic GR this would be curvature, but there are other possibilities. We do not actually know the set of all possible solutions to this problem; LQG was just one example, and whether or not the finally accepted solution will fall into one of the two categories mentioned by yourself remains yet to be seen. All of this is, at this point in time, very much pure speculation. We quite simply don't know yet how to reconcile GR and QM.

You see, here's the problem - if you consider a spherical shell of mass surrounding a ( spherical ) cavity, then Birkhoff's theorem implies that the geometry of space-time in the interior of the shell is actually Minkowskian. What that means is that, if you sit inside a hollow space surrounded by a shell of mass, you experience no gravitational time dilation at all as compared to an observer at infinity.

I am not certain what you mean here. I only mentioned LQG as an example; truth is, at this point in time we simply don't know which of our current hypothesis, if any, will represent a valid model of quantum gravity. No, you are right of course. I should have said that the idea of LQG is that macroscopically it becomes indistinguishable from GR, I did not mean to imply that it is actually the case. You are right that the proof of this is, as per yet, still outstanding. Presently LQG is really just a hypothesis with many ends still be tied together. My apologies for any confusion, I should have been more precise.

LQG is scale dependent; on macroscopic scales it is indistinguishable from deterministic GR, whereas on microscopic scales it is probabilistic in nature. So it incorporates both models. This works because on a macroscopic level probabilistic effects are so small as to be negligible.

Yeah, that's the big question, isn't it. One such attempt is Loop Quantum Gravity; basically what it boils down to is that on large scales it should be indistinguishable from standard GR ( proof of this is pending though ! ), whereas on small scales space-time itself becomes quantized, inducing quantum effects. This would be one way to do it.

It's neither of these. What is needed is an entirely new paradigm, which reduces to GR and QM in their respective energy domains, or implies these as boundary conditions. Consider, just as an example, String theory - the model is mathematically consistent only on a curved space-time background, and if you investigate the constraints on that curvature, you find precisely the GR field equations. At the same time, the energy levels of a String are ( roughly speaking ) its vibrational modes, which are inherently quantized. So there you have it - a model of quantized particle fields, which lives in a space-time governed by GR. P.S. I am not saying that String Theory is a valid model of quantum gravity; we don't know that. I am just using it as an example.

It is incomplete in the sense that it does not incorporate any quantum effects, it is a purely deterministic theory. For example, GR can model the gravitational collapse of a star only so long as quantum effects can be ignored; at the point of the collapse where such effects become important, GR ceases to be a useful model. That is why GR cannot tell us what really happens in the vicinity of a gravitational or cosmological singularity, or even if such singularities really do exist or not. To fully comprehend the physics of such a scenario we would need to have a model which incorporates both relativistic and quantum effects, namely a theory of quantum gravity. This is an area of active and ongoing research.

Division by zero is not defined; writing "1/0 = infinity" is meaningless.

It literally depends on your point of view. For an observer who is located far away outside the BH, an object falling towards the BH would take an infinite amount of the observer's own time to reach the event horizon, while slowly fading away into nothingness. On the other hand, if you were to "piggy back" on the infalling object you would not notice anything special. You would reach - and cross ! - the event horizon in a well defined, finite amount of time as measured on your own watch. Both observers are right, but only in their own frames of reference.

Of course, I completely agree. GR is a deterministic ( classical ) theory, whereas all of quantum physics is based on probabilistic principles. That is one of the reasons why they are so hard to reconcil / unify. We all understand that GR is incomplete in that its domain of applicability is limited.

This is good advice, right here

I disagree. The mathematics are very tedious ( as in time consuming ), but not complicated to understand conceptually. In fact the tensor relation which is the Einstein Field Equations is probably as simple and straightfoward as it gets, on a conceptual level. Turn on the GPS in your car - that's a beautiful demonstration of General Relativity right there. Without taking into account relativistic effects, that GPS would be off by something like 10km each day. A classical model that explains and quantifies the effects of gravity as being a geometric property of space-time, as opposed to mechanical forces between bodies.

It always amazes me how people go on and on, even though they have already been shown wrong. I will never understand this...

Fair enough You see, what is called the "twin paradox" in relativity textbooks & literature is a fairly specific setup with quite specific specs - one of them is the presence of acceleration for one of the twins. Yes, we can find other setups which lead to the same result even without acceleration ( which is what you were referring to ), but I truly believe that calling these alternatives "twin paradox" as well leads to a lot of confusion, as unfortunately happened on this thread. Anyway, I think it is all cleared up now. I am more of a "GR guy", but sometimes going back to the basics is interesting too.

Excellent, if we can all agree on this then we're good I was just starting to get confused as to what the actual argument was, since participants here appeared to be going in different directions, talking about different things.

I'm afraid I don't follow you. I never said anything about "preferred frames". In fact I explicitly stated that the frames are interchangeable without affecting the physical outcome, thereby ruling out that one of them is "preferred". That automatically makes them symmetric. I do not know why you state that on the one hand the motion is relative and symmetric, whereas the "situation" is not. That makes no sense. There is no such thing as "instant acceleration". I must admit that I am starting to wonder what the actual purpose of this thread really is. Of course it is possible to make up the elapsed proper time of an accelerating observer by combining the times of two purely inertial observers. So what ? That does not allow us to state that "acceleration is not important in the twin paradox". The two scenarios are not physically equivalent, in the sense that if you conduct them right next to each other, the participants will always be able to tell which experiment they take part in. So could somebody just state in plain text what the premise of the OP actually is ? I am getting more and more confused. So far as I am concerned acceleration is an integral part of the "twin paradox" scenario as it appears in relativity textbooks, because the point of the exercise is to teach students the physical difference between symmetric and non-symmetric frames. If you eliminate the asymmetry, you eliminate the discrepancy between the two twins after they are brought back together in the same frame at relative rest, defeating the purpose. Likewise, if you eliminate the "bring back together at rest in the same frame" bit, you depart from the original twin paradox scenario. Yes, precisely my point. So, the premise of the thread that "acceleration is not important in the twin paradox" is clearly false. You can consider alternatives to the typical twin paradox scenario to achieve the same outcome, but these are then different scenarios, which are not physically equivalent.