The CA in there was supposed to say CD.When we compare clocks CD and CL at the end of the experiment, we find that D > LA (which means that the ticking rate of CA > the average tick rate of CL for the whole trip. In the same way, we have E > SA (which means that the ticking rate of CE > the average tick rate of CS for the whole trip. There are only four clocks in my thought experiment.
Does Special Relativity contain contradictions?

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Re: Does Special Relativity contain contradictions?
Correction:

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Re: Does Special Relativity contain contradictions?
The reason for the "local" requirement is that real gravitational fields, such as that of the Earth, are not uniform. i.e to put it in vector calculus terms, the vector field that represents them has nonzero divergence through any closed surface containing the source of that field. Acceleration is equivalent to being in a uniform gravitational field. And being equivalent means that no experiment can distinguish between the two. That includes experiments that measure the relative tick rates of clocks.Halc wrote:The article you quoted misstates the principle of equivalence: "Einstein’s principle of equivalence tells us that whatever is true for acceleration is true for a gravitational field."
That is just wrong, so I question the rest of the article. The principle says there is no local test to determine which situation you're in.
The explanation I gave in my recent post explains how the principle discussed in the article about the accelerating rocket can be used for an infinitesimally small height (within which the gravitational field comes infinitesimally close to being uniform) integrated over a finite height.

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Re: Does Special Relativity contain contradictions?
Please see such posts as this:David Cooper wrote:You want both D>L1 and E>S1 to be true at the same time, but that is mathematically impossible (in set 2 or 3 models).
viewtopic.php?p=321015#p321015
to see what I "want".

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Re: Does Special Relativity contain contradictions?
Not good enough. If D>L1, it is impossible for E>S1.Steve3007 wrote: ↑October 7th, 2018, 5:46 pmPlease see such posts as this:David Cooper wrote:You want both D>L1 and E>S1 to be true at the same time, but that is mathematically impossible (in set 2 or 3 models).
viewtopic.php?p=321015#p321015
to see what I "want".
Frame CD analysis says that D is greater than L1, so if that's an accurate description of the underlying reality, then the claim from frame CE analysis which says that E is greater than S1 is an incorrect description of the underlying reality.
Alternatively, if frame CE analysis provides an accurate description of the underlying reality (when it says that E is agreater than S1), then the claim from frame CD analysis which says that D is greater than L1 is an incorrect description of the underlying reality.
The underlying reality can only be one way  not both at once. There is only one reality. You claim it can be both at once though (which is why I said that you want D>L1 and E>S1 at the same time), and in doing so you tolerate a contradiction.
(And if you object to the way I've described the claims generated from the CD or CE frame analysis on the basis that no such claims are being made, then you're forcing a change to wordings such as "Frame CD analysis says that D appears to be greater than L1", at which point you are recognising that the appearances can be deceptive and that the underlying reality is not always as it appears, so it doesn't get you any further  only one thing can be happening in the underlying reality, and it's either D>L1 or E>S1; not both.)
 Halc
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Re: Does Special Relativity contain contradictions?
SR doesn't demand the short duration accleration. It works fine with gradual acceleration, but that complicates the mathematics of what was a simple example. The simple example could be done with a tag team and no acceleration, with identical results, further evidence that the acceleration itself plays no role.Steve3007 wrote: ↑October 7th, 2018, 4:51 pmAs I explained, in order to switch between two inertial reference frames without spending finite (nonzero) time in a noninertial reference frame, the SR explanation uses the limit of a switch between those frames in a vanishingly small time period with an acceleration, during that time period, that tends to infinity.
SR is special not because of use of limits, but because of being a model of flat space.

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Re: Does Special Relativity contain contradictions?
If by "tag team" you mean more than one travelling observer then, yes, that's right. The simplest scenario is to imagine another traveler, travelling at speed v, who passes traveler B at C. Both B and C then remain in their respective inertial reference frames. But that's just the simplest scenario. We could have a series of inertial travelers all taking over the baton (as it were) in order to be able to consider the problem as a series of nonaccelerating, but not comoving, reference frames. Or, equivalently, we could imagine just the one traveler making a series of sudden changes of inertial reference frame. In other words, a series of changes, for each of which the time spent making the change tends to zero and the acceleration during that infinitesimal amount of time therefore tends to infinity.Halc wrote:SR doesn't demand the short duration acceleration. It works fine with gradual acceleration, but that complicates the mathematics of what was a simple example. The simple example could be done with a tag team and no acceleration, with identical results, further evidence that the acceleration itself plays no role.
Referring back to the Integral Calculus I mentioned earlier, this would effectively be a process of numerical integration. Remember, when you think of an arbitrary mathematical function y = F(x), the integral is the area under the graph. Numerical integration means finding that area by dividing it into columns. Integral Calculus means allowing the width of the columns to tend to zero. This is, essentially, what I was talking about earlier when I showed the outline of the mathematical process of going from the mathematics of Special Relativity to the mathematics of General Relativity.
As I've said before, I'm rusty on this. I haven't studied it for a long time. But I'm pretty sure the general idea is correct.
The idea of a model in which space is not flat, again, didn't come out of nowhere. It came out of a consideration of the implications of Special Relativity and the equivalence of inertial and gravitational mass, originating in Galileo's experiments. This led to the equivalence of accelerating reference frames and reference frames in the presence of a uniform gravitational field (or, in other words, the equivalence of accelerating reference frames and reference frames in a small, "local", region of a real, radial gravitational field, within which the variation of the field can be neglected, and it can therefore be regarded as locally uniform).SR is special not because of use of limits, but because of being a model of flat space.
This equivalence, together with considerations of the behaviour of light in an accelerating reference frame, is what led to the idea that light behaves in this same in a uniform gravitational field, and then to a consideration of the way that light behaves in a real gravitational field, and thereby to the idea that spacetime is not Euclidean. That's what was happening between 1905 and 1915.
So being "special" is not directly because spacetime is regarded as "flat". There is no direct intrinsic connection between the concepts of "specialness" and "flatness"!. In physics, being special means being a limiting case. That is, a case in which certain idealizing assumptions are made. i.e. a particular set of circumstances that the more general case doesn't require. In physics, the general contains the special. Special Relativity is the limiting case of General Relativity for the particular simplifying idealization of reference frames that are far away from any gravitating bodies and (equivalently) are not accelerating. Or (Einstein later realized) are in free fall.
Likewise, for example, classical mechanics is the limiting case of quantum mechanics for the special case in which momenta are large enough that such things as the uncertainty principle can be neglected. The EM wave model of light is a limiting case in which the number of photons is large enough that the EM wave equations (Maxwell's Equations) are good enough. Classical mechanics is the limiting case in which relative velocities are low enough that the Lorentz transformations can be neglected. etc.

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Re: Does Special Relativity contain contradictions?
Note: the more complete mathematics of General Relativity is more complex and involves Tensor Calculus, and the solutions to the equations are not exact, partly because of the way in which (in the GR model) mass causes spacetime to curve and spacetime causes mass to move. i.e. there's a feedback loop.

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Re: Does Special Relativity contain contradictions?
viewtopic.php?p=321292#p321292
Which particular part(s) or that post of mine is/are, in your view, "not good enough"?David Cooper wrote:Not good enough....Steve3007 wrote:Please see such posts as this:
viewtopic.php?p=321015#p321015...

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Re: Does Special Relativity contain contradictions?
Regarding this post:
viewtopic.php?p=321278#p321278
in which I tried to show the relationship between the Schwarzschild solution to Einstein's field equations for General Relativity and the time dilation in an accelerating rocket/room/elevator using Special Relativity, I was sufficiently interested to write a computer program which does the numerical integration. And it does indeed seem to numerically confirm that the two methods give the same result. So this confirms that the gravitational time dilation of General Relativity does indeed derive directly from consideration of Special Relativity and the equivalence of inertial and gravitational mass.
Interesting!
I'll post some screenshots of the output from the software some time in the next few days. If anyone is interested enough to try the software themselves, send me a private message with an email address and I'll be happy to send a copy of both the executable and the source code.
It's written in C# using Microsoft Visual Studio 2010.
viewtopic.php?p=321278#p321278
in which I tried to show the relationship between the Schwarzschild solution to Einstein's field equations for General Relativity and the time dilation in an accelerating rocket/room/elevator using Special Relativity, I was sufficiently interested to write a computer program which does the numerical integration. And it does indeed seem to numerically confirm that the two methods give the same result. So this confirms that the gravitational time dilation of General Relativity does indeed derive directly from consideration of Special Relativity and the equivalence of inertial and gravitational mass.
Interesting!
I'll post some screenshots of the output from the software some time in the next few days. If anyone is interested enough to try the software themselves, send me a private message with an email address and I'll be happy to send a copy of both the executable and the source code.
It's written in C# using Microsoft Visual Studio 2010.

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Re: Does Special Relativity contain contradictions?
I am not sure if this conflicts with Steve's calculations and I am not sure if this is clarifying or confusing, but anyway, another line of thought.
In the flat spacetime there are many inertial reference frames, and acceleration is needed for frame change.
In the curved spacetime there are many freefalling inertial reference frames, and acceleration is needed for frame change. But now there is a difference in gravitational potential between the top and bottom of the rocket during acceleration, and therefore the clock rates at the top and bottom differ. In free fall the clock rates are the same.
In the flat spacetime there is no gravitational potential, and therefore the clock rates are the same also during acceleration.
What do you think? Can it be as simple as this?
In the flat spacetime there are many inertial reference frames, and acceleration is needed for frame change.
In the curved spacetime there are many freefalling inertial reference frames, and acceleration is needed for frame change. But now there is a difference in gravitational potential between the top and bottom of the rocket during acceleration, and therefore the clock rates at the top and bottom differ. In free fall the clock rates are the same.
In the flat spacetime there is no gravitational potential, and therefore the clock rates are the same also during acceleration.
What do you think? Can it be as simple as this?
 Halc
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Re: Does Special Relativity contain contradictions?
Yes, that's exactly what SR does when considering the accelerating case. You do calculus.Steve3007 wrote: ↑October 8th, 2018, 2:16 amIf by "tag team" you mean more than one travelling observer then, yes, that's right. The simplest scenario is to imagine another traveler, travelling at speed v, who passes traveler B at C. Both B and C then remain in their respective inertial reference frames. But that's just the simplest scenario. We could have a series of inertial travelers all taking over the baton (as it were) in order to be able to consider the problem as a series of nonaccelerating, but not comoving, reference frames.
The point seems to be that the acceleration itself causes no dilation. My two clocks (equator and in northpole centrifuge) should run at identical rates. Everything is the same for both of them except the acceleration.
Gravitational force also does not cause dilation. I have a clock here on Earth, and one on my lab on Uranus where I weigh about 8/9th of Earth. But the Uranus clock is dilated more, despite the weaker force of gravity. It feels like less acceleration, yet the dilation effect is more. Clearly the acceleration, or a gravitational field that feels like acceleration, plays no direct role in the dilation. It is caused by the negative gravitational potential, which is far greater on Uranus (about 3x Earth), and nonexistent in a centrifuge.
Doesn't work. Try it with my Uranus example, where the dilation factor is not a function of the equivalent acceleration. GR is not derived directly from SR in this way. Dilation is apparently proportional to (equal to?) that of a point's escape velocity, not equivalent acceleration. The calculus leading to GR may have been worked out from SR in that manner.This is, essentially, what I was talking about earlier when I showed the outline of the mathematical process of going from the mathematics of Special Relativity to the mathematics of General Relativity.
There is no concept of escape velocity from an accelerating rocket. No matter how fast you fire an object out of the top of the rocket, the rocket at any nonzero acceleration will eventually catch up to it.
Equivalence says you can't tell the difference between the two with a local test. It does not imply any dilation, which is not locally detectable. Equivalence principle does not suggest that time is dilated due to acceleration.This led to the equivalence of accelerating reference frames and reference frames in the presence of a uniform gravitational field (or, in other words, the equivalence of accelerating reference frames and reference frames in a small, "local", region of a real, radial gravitational field, within which the variation of the field can be neglected, and it can therefore be regarded as locally uniform).

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Re: Does Special Relativity contain contradictions?
Gravitational time dilation very much is a function of acceleration due to gravity (g) and it is also a function of escape velocity v_{e}. v_{e} is a function of g. See equation 5 below.Halc wrote:Doesn't work. Try it with my Uranus example, where the dilation factor is not a function of the equivalent acceleration. GR is not derived directly from SR in this way. Dilation is apparently proportional to (equal to?) that of a point's escape velocity, not equivalent acceleration. The calculus leading to GR may have been worked out from SR in that manner.
There is no concept of escape velocity from an accelerating rocket. No matter how fast you fire an object out of the top of the rocket, the rocket at any nonzero acceleration will eventually catch up to it.
As I said in this post:
viewtopic.php?p=321278#p321278
The time dilation for a point at distance 'r' from the centre of a gravitating body, with respect to a point an infinite distance from the centre of the gravitating body, is given by the Schwarzschild solution to Einstein's field equations from General Relativity:
1.
t_{0} = t_{f}(1  (2GM / rc^{2}))^{0.5}
The acceleration due to gravity at point 'r' is given by:
2.
g = GM / r^{2}
Therefore, substituting 1 into to 2 gives:
t_{0} = t_{f}(1  (2gr / c^{2}))^{0.5}
So the time dilation is  a function of 'g'  the acceleration due to gravity.
The escape velocity is given by:
3.
v_{e} = (2GM / r)^{0.5}
Therefore, substituting 3 into to gives:
4.
t_{0} = t_{f}(1  (v_{e}^{2} / c^{2}))^{0.5}
So, you see the time dilation with respect to a point outside the gravitational field (i.e. an infinite distance from the centre of the gravitating body) is indeed related to escape velocity. But that doesn't mean that it isn't related to acceleration due to gravity.
From 2 and 3:
5.
v_{e} = (2gr)^{0.5}
As I've said, the reason why the equivalence principle uses the word "local" is that acceleration is equivalent to a uniform gravitational field.Equivalence says you can't tell the difference between the two with a local test. It does not imply any dilation, which is not locally detectable. Equivalence principle does not suggest that time is dilated due to acceleration.
Steve3007 wrote:The reason for the "local" requirement is that real gravitational fields, such as that of the Earth, are not uniform. i.e to put it in vector calculus terms, the vector field that represents them has nonzero divergence through any closed surface containing the source of that field. Acceleration is equivalent to being in a uniform gravitational field. And being equivalent means that no experiment can distinguish between the two. That includes experiments that measure the relative tick rates of clocks.
etc.Steve3007 wrote:This led to the equivalence of accelerating reference frames and reference frames in the presence of a uniform gravitational field (or, in other words, the equivalence of accelerating reference frames and reference frames in a small, "local", region of a real, radial gravitational field, within which the variation of the field can be neglected, and it can therefore be regarded as locally uniform).
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Re: Does Special Relativity contain contradictions?
I think our disagreement is a language one.
Let us consider a rocket 300 km tall, sitting on nonspinning Earth. The clocks at top and bottom are synced at time zero, but are an entire millisecond apart, so each observer looking at the other sees 0.001 seconds on the opposite clock.
After years, due to the difference in gravitational potential, Alice's clock gets a minute ahead of Bob's clock, which is verified when they look at each other's clocks, still with that 0.001 second discrepancy.
Same rocket now, but accelerating in space. At time zero, the clocks are synced and both see 0.001s on the other clock. My description is initially done in this inertial frame.
This is where our language departs, because equivalency principle talks about accelerated frames, so it doesn't seem appropriate for me to talk about it only in the inertial frame.
So in that inertial frame, years go by and the rocket is going really fast. Bob actually moves a little faster and is much closer to Alice now, but his clock has fallen about a minute behind in the original inertial frame due to the effort required to catch up like that.
Light from Alice's clock takes well under a millisecond to go down to meet Bob coming up the other way, so Bob sees Alice's clock about one minute fast. Light from Bob on the other hand is barely moving faster than Bob and has to slowly catch up to Alice at the top, which takes 2 minutes, so Alice sees a 1minute lag in Bob's clock.
I didn't do the math. Notice a lack of the amount of time it would take to get Alice's and Bob's clocks to be off by an hour at 1G. But equivalence principle holds. Alice and Bob even after their clocks get off by a minute cannot tell if their rocket is on the ground or is accelerating in space.
That's the same thing both see in the gravitational field, just viewed from an inertial frame. But it is their speed in this instance that causes their discrepancy in the inertial frame, not the acceleration.
The centrifuge thing eliminates all other factors except the acceleration, and notes no dilation difference. The Uranus thing holds equivalent acceleration constant, and yields very different dilation. Both are evidence that acceleration is not itself the cause of the dilation. That's what I've been trying to say.
Let us consider a rocket 300 km tall, sitting on nonspinning Earth. The clocks at top and bottom are synced at time zero, but are an entire millisecond apart, so each observer looking at the other sees 0.001 seconds on the opposite clock.
After years, due to the difference in gravitational potential, Alice's clock gets a minute ahead of Bob's clock, which is verified when they look at each other's clocks, still with that 0.001 second discrepancy.
Same rocket now, but accelerating in space. At time zero, the clocks are synced and both see 0.001s on the other clock. My description is initially done in this inertial frame.
This is where our language departs, because equivalency principle talks about accelerated frames, so it doesn't seem appropriate for me to talk about it only in the inertial frame.
So in that inertial frame, years go by and the rocket is going really fast. Bob actually moves a little faster and is much closer to Alice now, but his clock has fallen about a minute behind in the original inertial frame due to the effort required to catch up like that.
Light from Alice's clock takes well under a millisecond to go down to meet Bob coming up the other way, so Bob sees Alice's clock about one minute fast. Light from Bob on the other hand is barely moving faster than Bob and has to slowly catch up to Alice at the top, which takes 2 minutes, so Alice sees a 1minute lag in Bob's clock.
I didn't do the math. Notice a lack of the amount of time it would take to get Alice's and Bob's clocks to be off by an hour at 1G. But equivalence principle holds. Alice and Bob even after their clocks get off by a minute cannot tell if their rocket is on the ground or is accelerating in space.
That's the same thing both see in the gravitational field, just viewed from an inertial frame. But it is their speed in this instance that causes their discrepancy in the inertial frame, not the acceleration.
The centrifuge thing eliminates all other factors except the acceleration, and notes no dilation difference. The Uranus thing holds equivalent acceleration constant, and yields very different dilation. Both are evidence that acceleration is not itself the cause of the dilation. That's what I've been trying to say.
 Halc
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Re: Does Special Relativity contain contradictions?
That's a function of g and r. I cannot determine my dilation if I only know g. A rocket doesn't have an r. Uranus has only a little less g (I selected it for that reason. Saturn is closer to Earth g, but it's a little more), but has much more r, so the escape velocity and the dilation is about 3 times as much.
I think you are talking about relative dilation (that between Alice and Bob), and I'm talking about absolute dilation (between Alice and somebody not accelerating, or between Alice on Earth and somebody not on Earth. Locally, Alice and Bob only have access to relative dilation, and Alice's clock is going to get ahead of Bob's clock from their POV in all cases. I'm not denying that. But how much their clock is dilated compared to somebody not dilated at all is not a function of acceleration or g (same thing). The Uranus observer is going to run slower despite the lower g, compared to the Earth observer. Alice and Bob on Uranus will diverge more slowly there than they would on a building on Earth, since that divergence is a function of g, even if their actual dilation is not.
Yes, gravitational dilation is a function of escape velocity (although the relative dilation you're talking about is not). I can determine the actual dilation from only that figure, even without knowing g. There is no local test to determine escape velocity. A rocket doesn't have an escape velocity.
That equates to setting r to infinity, which is very much like the situation with the rocket. But that value also sets the dilation to stoppedtime (moving at c). But yes, it is considered cheating (a nonlocal test) for Alice to measure less weight up there and conclude she must be in a gravitational field.As I've said, the reason why the equivalence principle uses the word "local" is that acceleration is equivalent to a uniform gravitational field.
My example just posted above probably should have had Alice and Bob in separate rockets, one ahead of the other. That would get rid of some funny comparisons about when Alice starts to move relative to the event of the engines kicking in a Bob's end. It removes the length contractions that messed up my calculations. The two remain 300km apart for the duration, at least in the original inertial frame, but they grow aopa

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Re: Does Special Relativity contain contradictions?
Where does it address the issue? It fails to break the proof that set 2 models generate contradictions, and you fail to admit that set 2 models generate contradictions (that rule them out). In models with running time (i.e. not set zero's magic static blocks) and where clocks don't all tick at the same rate at all times (i.e. set 1's models with eventmeshing failures), contradictions are generated unless you accept an absolute frame. I proved the point, but you refuse to accept that, and all you can do now is link to irrelevance to cover up the fact that you lost the argument.Steve3007 wrote: ↑October 8th, 2018, 2:31 amviewtopic.php?p=321292#p321292Which particular part(s) or that post of mine is/are, in your view, "not good enough"?David Cooper wrote:Not good enough....