Aha. In the few minutes since I posted reply #60 just above, I’ve also read this. This is a very readable explanation, such as I have not seen before. (Now I really do need to go read the series of pages about Twin Paradox, cited earlier.)
Is it fair to argue that there are, in fact, not just three frames of reference, but infinitely many at work here? That is, since the ship is accelerating/decelerating continuously and thus always changing its instantaneous velocity, does this mean that each moment of distinct velocity is a different frame of reference than each other moment?
May I go out on a limb here and ask a question about relativity, in total ignorance: Is the above point, in fact, exactly what GR deals with that SR does not?
Yes, and no. Special relativity is perfectly capable of dealing with continuous acceleration (and hence, continuously-changing reference frames). It’s a little more complicated, of course, in that you have to bring in calculus, but that’s the only additional math you need to bring in.
Einstein’s example was not based solely on the Earth. The man in the room was on the surface of the Earth, but the man in the chest was being accelerated “by a hyothetical being pulling on a rope…”. Also, before the experiment starts, Einstein places the man in the chest “…in a large portion of empty space…” and says “Gravity naturally does not exist for this observer.” If we are to believe Einstein that gravity and acceleration are indistinguishable, then the clocks must run at the same rate, or they would not be indistinguishable. I totally agree with Keeve that in freefall, or under any acceleration other than 1G, the clocks will not stay in sync, but these 2 clocks are both in 1G, and therefore will stay in sync. Keeve’s argument ( and any argument along the same lines) is therefore flawed.
I am of course aware that mainstream science says that nothing can travel faster than c, and I covered that with the Lorentz equations concerning mass increase. Perhaps I did not go into enough detail, but paragraph 3 covers it as far as I can see. Maybe I was being a bit theatrical in the opening sentence, for that I apologise.
If anyone disagrees that 2 clock both under 1G will not run at the same rate, would you please explain why.
What you are missing is that it is not the force of gravity that determines how fast the clocks run. It is how far down the gravitational well that matters.
ETA: You could do worse than to read the Wikipedia article. Here’s the very first sentence:
You are correct if you are assuming one grativational well, and 2 clocks - which may or may not be at the same level. The 2 clocks at the same level will run at the same rate, the 2 clocks which are not, will not.
If the 2 clocks are both experiencing 1G, whether they are in the same well or not, will run at the same rate (and therefore stay in sync).
Here are the first 2 paragraphs under the heading “Definition” from your link
“Clocks which are far from massive bodies (or at higher gravitational potentials) run faster, and clocks close to massive bodies (or at lower gravitational potentials) run slower (slow is low). This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects.
It can also be manifested by any other kind of accelerated reference frame such as an accelerating dragster or space shuttle…”
The second paragraph is relevant.
You are bantering semantics instead of answering my point. I could be in the room on the surface of the Earth, or in the chest being accelerated at 1G, and I would not know which, as in both cases, I would experience 1G. The clocks in the room and chest are both experiencing 1G, whether this is due to gravitational potential or force or acceleration is immaterial. Subsitute “hypothetical being pulling on a rope” with internal rocket motor, and you have my scenario.
As requested earlier, please supply proof that the clocks run at different rates.
It is most certainly not immaterial. Two clocks experiencing 1G could be at the same gravitational potential, in which case they will stay in sync. They could be at very different gravitational potentials, in which case they will not stay in sync.
I’ve already provided a link that gravitational time dilation is due to difference gravitational potential.
You keep glossing over the difference between gravitational potential and gravitational force, which tells me that you don’t understand the difference. You apparently think it is just “semantics”. Again, learn the difference.
The problem is that neither graviational force nor graviational potential are really basic properties of gravity in general relativity, instead they emerge when you impose (the same) restrictive conditions on spacetime.
The big problem with comparing clocks in relativity is that different procedures can lead to different results and in general relatvity there is no obvious general procedure for comparing clocks other than if the clocks start and finish at the same events. A lot of the time comparing clocks of different observers is subjective.
The truth is that both the non-geodesic nature of a test particle’s worldline ‘held’ in a graviational field (read: the gravitational force it experinces) and the nature of the gravaitional field it is in will affect the proper time that the test particle experinces. Just as the ‘straightness’ of a walkers path and the ‘bumpiness’ of the terrain will both affect the length of a the walkers paths between two points.
In the simplest examples both the gravitional force and the gravaitional potential are functions of each other, so which is the culprit behind gravtional time dialtion is moot. In more complicated examples a reasonably non-subjective way of comapring clocks becomes more difficult to find or notions of gravitational force and graviational potential become more difficult to define.
I have bantered semantics with you long enough. I do know the difference. Put succinctly, gravitational potential exists everywhere, even for a body in freefall. Gravitational force, however, only exists for a body which is prevented from responding to gravity. For example a body on the Earth, or a body undergoing 1G acceleration.
If your 2 bodies (clocks) are in the same gravity well, ie the same gravitational potential, but at different distances from the gravitating body, they will not be both at 1G and therefore not stay in sync. If they are the same distance, they will both experience 1G (at the correct distance of course) and stay in sync. They could be in 2 different gravitational potentials, but positioned so they both experience 1G, in which case they stay in sync; or positioned so one is experiencing 1G and one is not, in which case they will not stay in sync.
Instead of just waving your hands and asserting they do not stay in sync at 1G, please supply proof. Bear in mind that gravitational mass and inertial mass are equivalent, I refer you again to chapter XX of Einstein’s book.
This is the Wiki passage which you are no doubt quoting from :- “Clocks which are far from massive bodies (or at higher gravitational potentials) run faster, and clocks close to massive bodies (or at lower gravitational potentials) run slower (slow is low). This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects.”
This is ambigious, as it does not actually say what state of motion these clocks are in. The assumption (my assumption anyway), is that they are at rest relative to the “massive bodies”, because if they were in freefall, they would both be experiencing zero G. Going back to the equivalence principle, the man in the chest could tell whether he was in a gravitational field or was being accelerated. The two systems would not be equivalent if the clocks ran at different rates.
I invite comments from other interested parties with a point of view.
Read that Wikipedia line again: "the lower the gravitational potential, the more slowly time passes. Your statement is plainly in contradiction with that.
Wrong. I’m not going to try to explain the difference to you, because I’d be wasting my time, but what you wrote is wrong.
There, I’ve bolded where your problem is. Stop assuming. Read what it says: “Clocks […] at higher gravitational potentials) run faster, and clocks […] at lower gravitational potentials) run slower”. Also pay attention to what it doesn’t say: anything about having to be in freefall or not, anything about being at rest or not, anything about gravitational force affecting clocks.
Good, because I’m done here. It’s your right to be ignorant, if that’s what you want.
Your both right and wrong to various degrees here. The Wikipedia article is a tad misleading in places, but you’ll note that it gives the formula for graviational time dilation for an observer held ‘stationery’ in the vacuum of a spherical gravitational field here and the formula for the gravitational time dilation of a free-falling obsever in a circular orbit in the same graviatiional field here.
Note that the free-falling observer does experince gravitational time dilation, but that the two formulae are not the same. So in fact gravitational time dilation would appear to be both dependent on the ‘gravitational force’ experinced by an observer and the ‘gravitational potential’ of the observer.
One thing I can’t stress enough though is that this is entirely dependent on how you choose to compare clocks and both the formulae above are for a highly symmetric spacetime (Schwarzchild spacetime) which lends itself to making such comparsions. Not all spacetimes are highly symmetric and asympotically flat (i.e. have faraway observers which for practical purposes do not experince gravtiational effects for which we can compare clocks with) and so concepts such as gravitational force, gravitational potential and gravitational time dilation become more difficult to pin down.
The difference doesn’t come from the difference in force, it comes from the difference in velocity. From here (PDF), on page 3:
(dTau/dt)^2 = (1 - 2M / r) - v^2 (In units where G == 1)
For the circular orbit, the term v^2 contributes the extra **-M/r **quantity.
This means for a third observer in freefall, initially rising and with his peak at the same radius r, when he reaches the peak and is stationary, v^2 = 0, and his time dilation will be the same as that of the the stationary observer, even though the stationary observer is experiencing force, and the third observer is in freefall. If the difference were due to force, you’d instead expect the third observer to have a time dilation matching that of the person in circular orbit.
And for a test particle of unit mass v[sup]2[/sup] term with a simple function of the net force ‘felt’ by the particle (and the variables M and r which already appear in the equation). So saying it’s the velcoity not the force that’s causing the time dilation is arbitary as anyone could say that it’s the force experinced by the observer that’s causing the circular movement with constant velcoity v.
The 3rd observer is only at r instantaneously, whilst it’s not wrong or even unhelpful to assign it a ‘time dilation factor’ equal to the hovering observer, it’s still arbitary as we haven’t even decided on a way to compare clocks for the 3rd observer.
I don’t really understand what you’re trying to say here. v is only a function of force because you’ve restricted the particle to a circular orbit. In general, v and the force are independent.
We can choose a short time span, so that v^2 is negligible.
Look, I’ve already provided cites that gravitational time dilation depends on gravitational potential. Here’s another one. Note that there are potential and velocity terms, and no force terms. In fact, do a search for the word “force” on that whole Time Dilation page. How many times is it there? Zero.
Can you (or anyone) provide an actual cite that gravitational time dilation depends on force?
[QUOTE=Clocks which are far from massive bodies (or at higher gravitational potentials) run faster, and clocks close to massive bodies (or at lower gravitational potentials) run slower (slow is low). This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects.[/QUOTE]
I still maintain that the above paragraph is ambigious, as so much on this subject is. The phrase “far from massive bodies”, makes no mention of being in freefall or undergoing movement of any sort. If I say to you "I am far from New York, I have given you no indication at all of my state of motion. ZenBeam has asserted that my assumption that the clock is stationary [in this gravitational field] is wrong. But he is also making an assumption; his assumption is that the clock is in freefall. I assert that his assumption is wrong.
I posted previously :- “They [the clocks] could be in 2 different gravitational potentials, but positioned so they both experience 1G, in which case they stay in sync.” To which you replied :-
Read that Wikipedia line again: "the lower the gravitational potential, the more slowly time passes. Your statement is plainly in contradiction with that.
My statement does not contradict that Wikipedia line. In any gravitational potential, provided of course it is large enough to cause an acceleration of more than 1G, a position can be found where, when the clock is prevented from falling towards the gravitating body, (ie held still), an accelerometer placed with it would read 1G. If the body was the mass of the Earth, the 1G acceleration point (sphere) would be at 4000 miles aprox. If the body was more massive, it would be closer, less massive, further. That in no way is a contradiction.
[QUOTE=tomh4040;14408219
I posted previously :- “They [the clocks]
could be in 2 different gravitational potentials, but positioned so they both experience 1G, in which case they stay in sync.”
[/QUOTE]
this is false, one of the confirming experiments of general relativity was to load a pair of super precise atomic clocks onto a pair of jets and have them fly in opposite directions relative to the earths spin (not certain if that was important or not) basically one is flying much faster than the other as a result and the clocks registered a difference on landing that was well within predictions laid out by GR.
they were in the same gravity. its speed AND gravity that have the effects of time dilation, well gravity does it all by itself and so does speed.