The problem is precisely that general relatvity is a metric theory of gravity.
Take the stress-energy tensor, this is a rank-2 tensor field which among other things describes the (non-gravitational) enegry of spacetime. In a vacuum this vanishes, which is precisely what we’d expect as a vacuum is a region that contains no non-gravitational energy.
So far no problems, but let’s imagine what sort of object could describe the gravaitional potential energy. By the equivalance principle, you can always choose a local frame (descrbing an observer in free fall) so that spacetime appears locally flat. Now a flat spacetime is one without an sort of gravitational energy, so locally the gravitational energy must also vanish. I.e. in gravitational energy, should it exist in general relatvity cannot be a local property of spacetime as it can be made to vanish locally just like the stress-energy tensor vanishes when describing a region containing no energy.
Already we’re at odds with the idea of gravitational potential energy, because the precise point of GPE is to give gravitational energy a local definition on par with other kinds of energy.
So you cannot derive a cocnept of gravitational potential energy purely from the general relativstic description of gravity, you need either explicitly or implicitly add something else such as a preferred frame of reference or a decompostion of the metric in to a gravitational part and a background part (this is driectly paraphrased from Wald’s General Relativity).
There are what are known as pseudo-tensor fields which can be defined on spacetime to allow us to find some sort of expression the total energy/energy flux (including graviational energy) of some arbitary volume of space, but that’s still not the same as GPE as they can be made to vanish locally (whereas in Newtonian gravity GPE does not vanish).
I believe the statement that time dialtion due to GPE is a central prediction of metric theories of gravity comes from MWT’s Gravity where it is shown that in the limit relativistic theories of gravity must reduce to a certain metric described by the Newtonian potential, from which graviational time dialtion can easily be related to gravitational potnetial.
The reason I think it’s too prescriptive to rule out the role of force is that in general relatvity terms force-like terns appear as Christoffel symbols related to some coordinate system and I think it’s just as easy to see time dialtion as a result of the aggregate effect they have on the transport of vectors in spacetime.
That makes sense, but we’re not really talking locally when we’re talking about gravitational time dilation - we’re talking about two different places in curved space-time. If we pick a frame that makes one of those places flat, then that frame makes the other one non-flat, and thus at a different potential, no?
Even in Newtonian gravity, you can assign the point where GPE is zero arbitrarily - two common choices are zero at infinity (which is helpful when talking about escape from a gravity well), and zero at “ground level” which is useful when distances are small.
Ok. That makes sense - I’m a firm believer that it’s useful to learn to address physics problems using a variety of different approaches, because sooner or later, you’ll run into problems that are easy with approach ‘a’ and difficult with approach ‘b’, and other problems that are easy with ‘b’ and hard with ‘a.’
Yes, graviational time dilation is an effect that comes from essentially not local. However; no, picking a local frame will not fix the frame at some other point, that I guess would be equivalent to saying that defining a local basis defines a tetrad basis for the whole of spacetime, which is not true and I’m not sure that there’s any sensible conditions you could impose on a tetrad bsis to make it true.
[/quote]
The issue you’ve brought up is gauge arbitariness and that’s not really the issue here. I feel a bit over my head discussing gauge theory aspects of general relativity, but the problem is not fixing a gauge for GPE (i.e. in particualr fixing a zero point)
Perhaps the best way to understand it is to compare to an elctrostatic field in general relativity. In an electrovacuum (i.e. a ‘vacuum’ containing only an elctromagnetic field), the stress-energy tensor does not vanish, unlike a ‘true vacuum’ in general relativity (i.e. one not containing any gravitational sources, though it still may contain a gravitational field) where it does vanish identically. Your just never goign to get a definition of graviational potential energy in general relativity where there’s a basic equivalnce it and for example electrical potential energy. Electromagentic energy exists locally in general relatvity, graviataional energy does not.
I’m not saying my description from my last post is better (it is in fact very hand wave-y), btu key to recognise is that in this situation there’s so much more going on in the general relativistic description of gravitational time dialtion-like effects than gravitational potential energy making clocks run slower.
Hi, sorry for my enforced absence, I’m joining in again.
I got so carried away with the ongoing arguments about 1G gravity versus 1G acceleration that I forgot my original mission. I was wrong in pursuing the argument about the 1G accelerated frame not going out of sync with the reference clock as I had the wrong parameters. Here are the correct parameters, and why I have been keen to establish the time connection between 1G gravity and 1G acceleration.
There is a closed chest resting on the surface of the Earth. Naturally the man inside experiences 1G. During the course of this experiment he is not allowed outside the chest.
There is a closed chest with a man inside in a rocket ship which is about to be accelerated at 1G. During the course of this experiment he is not allowed outside the chest.
The Earth and the rocket to be accelerated are both placed in intergalactic space, far away from any other bodies so there are no reference points except each other. They are aware of each other, and know they are eg 1 million Km apart.
Their clocks are synchronised and the rocket motor is fired up. The distance between the two then rapidly increases at 9.8m/s^2. The man in the rocket (he does not know he is in a rocket remember) experiences 1G, and time for him passes normally - ie he cannot detect any change in the passage of time. He can also conduct experiments which tell him he is experiencing 1G, and all experiments he conducts behave normally, just as they did when he performed them on Earth before this experiment began. The man in the chest on Earth experiences 1G and for him the passage of time is “normal” also. He can also conduct experiments which tell him he is experiencing 1G, and all experiments he conducts behave normally, just as they did previously on Earth before this experiment began.
After 2,191.5 hours by the clock on the rocket, the motor on the rocket is cut, the rocket is rotated round 180 degrees, and the motor restarted. The man in the chest on board does not know this, all he knows is that he was weightless for 60 seconds, then all was normal again. Unknown to the man in the rocket, 2,191.5 hours after that period of weightlessness, the rocket very briefly (almost instantaneously) stops (relative to Earth) and starts accelerating back. 4,383 hours after the first period of weightlessness, the procedure is repeated. All he knows is he was weightless again for 60 seconds. Another 2,191.5 hours passes, and the Earth and the rocket are adjacent to each other again. One year has passed on the rocket. The experiment has now ended, they are back again to the 1 million Km apart. The rocket cuts its motor and the clocks are compared. Note that the man in the chest on Earth did not undergo two 60 seconds periods of weightlessness, and this will cause a disparity between the clocks of 0.00008 seconds, a trifling amount.
With the above scenario, any GR effects are present in both FRs, and therefore cancel each other out. Are the clocks in sync to within a few seconds. If not, which clock is slow, and why?
We have here of course the twin paradox, as each man can justifiably say that he was the one who was stationary, and the other was moving. Their only reference was each other. They both experienced 1G for one year (except for one man who experienced 0G for 120 seconds, wich will not affect the outcome.
It can be arranged for the man in the chest on Earth to be weightless also for 2 periods of 60 seconds, by having him suspended and dropped, but the experiment is simpler as laid out here, and just as accurate.
You mean all he knows is that he was weightless for an instant, then accelerated sideways for almost a minute, then decelerated suddenly (to stop the rotation), then weightless for another instant, and then under 1G again. I think he’d notice that. Even if he was at the center of the ship, he’d notice the ship rotating around him, during the weightless period.
I think the length of the OP proves without a doubt the existence of relativity - to the OP, “now” means" after I write 15 paragraphs about something that I’m unsure of, which will make time move even more slowly outside of my mind because it is engaged and (pre?)occupied; to the guests, who by the end of the post have been ringing the doorbell for 30 minutes whilst freezing their asses off, those thirty minutes seemed like at least two hours.
The article does make an unsourced claim that Einstein said it to his publishers, but there’s no evidence for it. And he would certainly know it wasn’t true.
There’s a lot wrong with Tom’s assumptions, but the best way (IMO) to view it is as an unnecessarily overcomplicated version of the standard special relativstic twin parados, but with one (or both - it’s not clear from Tom’s post) being subject to a small peturbation due to general relativistic effects.
What’s clear is that the time dilation ‘suffered’ by the rocket ship observer massively outweighs the tiny peturbation caused by general relativstic effects and the rocket ship observer will experince much less time between the start and the end of their journey than is measured on the Earth bound observer’s clock.
The rocket is going at about 0.25c at that point, so if it ‘almost instantaneously’ stops, its passenger will notice… If only for a fraction of a second, after which he will have ceased noticing things for good.
But it’s easy to see that of course, the travelling clock will show less time elapsed than the stationary clock on Earth. As I said, after 2191.5 hours at 1g acceleration, the rocket is going at roughly 0.25c, meaning a gamma factor of roughly 1.03 – small, but noticeable. By contrast, the equivalent factor for the observer in the gravity well, giving the slowing down of his time relatively to a ‘far away’ observer, is roughly 1.0000000007, utterly negligible in comparison. As These are my own pants said, we can thus just think of this as the familiar twin problem, where of course less time passes for the moving observer.
No, what he means is that the rocket starts at zero velocity relative to Earth, then the rocket accelerates at 1g for 90 days, achieving .25c relative to Earth, then flips and accelerates in the opposite direction at 1g for another 90 days, at which point it will have zero velocity relative to Earth. The rocket keeps accelerating at 1g for another 90 days until it is back to .25c relative to Earth, then flips accelerates in the opposite direction at 1g for another 90 days, at which point it will have zero velocity relative to Earth.
At no point does the rocket instantaneously go from .25c to 0. The instantaneous point where the rocket changes from moving away from Earth to moving towards Earth won’t be felt at all in the rocket, because it is accelerating smoothly at 1g the whole time.
I think a part of Tom’s intention is to prove there are no special relativistic effects because it’s impossible for the two observers to know which is which, but that might just be my interpretation of his first post where he argues against FTL reversing the flow of time.
There’s also the issue of him considering 0.000008 s to be a trifling amount when doing GR calculations for 1G.
Yep, though of course as AndyL pointed out the two observers can distinguish each other due to the change in force. Not that that makes a huges difference in general relativity as it’s perfectly possible to have two observers who cannot peform any local experiment to distinguish themselves from each other, but who measure different times between events at which they are both present.
One thing to note is that I didn’t calculate the amount of time the rocket ship twin experinced,so when I said he will experince much less time that was a slight exaggeration (though to make iso its no exaggeration a all we need to do is to extend the time that the rocket accelerates) , however it is still painfully clear that the special relativstic twin effect dwarves the peturbation due to the Earth’s gravity.
0.000008 secs is a trifling amount, when considering that the experiment lasts for one year. The main thrust of the argument seems to have been missed. These 2 observers have no reference other than each other, so which one “actually” moved? Each man can justifiably claim that he was the stationary one. I have used GRT in this twin paradox problem to remove one of the arguments which is often heard when “proving” which twin actually moves. As both are under the same 1G, except for 120 seconds for one man, their situations are identical. They have no reference to the what can be called the standard clock far away from any gravitational source. As stated in my last posting, it can be arranged for the man in the chest on Earth to be weightless also for 2 periods of 60 seconds, by having him suspended and dropped (starting at 34Km above the Earth, dropping for 60 seconds, which leaves him at 17Km, then again, which puts him on the surface. This neglects air resistance of course, and three calculations have to be done, not just one. This just leaves the SRT time dilation effect, and the question, wich one moved and which one was stationary? is still not answered.
Okay we can both agree the question of which one “moved” is totally subjective and metaphysical, however the question we want to ask is what interval of time does each one measure between events which, whilst in the scenario you’ve set up there is an elemtn of subjectivity (which we will ignore), is an objective physical question
Okay, what your trying to do is to create some sort of symmetry between the observers, you haven’t managed that, but see below.
I think we can ignore air resistance without missing anything of qualitive importance.
As has been pointed out the rocket twin changes direction. In the special relativistic version, changing direction breaks the symmetry between the two twins and remember that special relativity is just a special case of GR.
However, try this scenario: after a period the twin on Earth drops through a hole which goes directly through the Earth’s centre and where he comes out the other side and stops. This exactly simulates what the rocket bound twin feels when they stop and then turnaround and whilst it may have a significant effect on the GR correction, whatever effect it does have will still be much smaller than the time dilation effect from the standard twin paradox. We can alos adjust the set-up in other ways.
So now we do have two apparently symmetrical situations, however there will still be a time dilation like effect between the two twins. The reason is that there is a global asymmetry is in the spacetime which both twins occupy.
Compare to the cosmological twin paradox of GR. In this both twins start and finish at the same spot and neither twin is subject to acceleration and the spacetime they occupy is (or at least can be made) very symmetric, yet there is a significant time dilation effect due to the asymmetry introduced by something as esoteric as them having different winding numbers in spacetime.
