Gravities Mechanism

The main point to get from Special Relativity's so-called Twin Paradox, is that imaginary elapsed coordinate time, Δt, is different than physically relevant elapsed proper time, Δτ.

Coordinate time is dependent on an imaginary choice of inertial coordinates, so varies from frame to frame by the Lorentz transform.
\(\Delta \vec{x}'_{AB} = \Delta \vec{x}_{AB} + \frac{1}{\vec{u}^2} \left( \frac{1}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}} - 1 \right) \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right) \vec{u} + \frac{\vec{u}}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}}\Delta t_{AB} \\ \Delta t'_{AB} = \frac{1}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}}\Delta t_{AB} + \frac{1}{c^2 \sqrt{1 - \frac{\vec{u}^2}{c^2}}} \vec{u} \cdot \Delta \vec{x}_{AB} \)

Proper time, while path-dependent, is independent of the choice of inertial coordinates use to describe a time-like trajectory through space-time:
\(\Delta \tau_{AB} = \int_{A}^{B} d\tau = \int_{A}^{B} \sqrt{1 - \frac{\left( \vec{v}(t) \right)^2}{c^2}} dt = \int_{A}^{B} \sqrt{1 - \frac{\left( \vec{v}'( t' ) \right)^2}{c^2}} dt'\)

This follows because the relativistic interval likewise is independent of the choice of inertial coordinates use to describe a time-like separation in space-time:

\( c^2 \left( \Delta t'_{AB} \right)^2 - \left( \Delta \vec{x}'_{AB} \right)^2 \\ = c^2 \left( \frac{1}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}}\Delta t_{AB} + \frac{1}{c^2 \sqrt{1 - \frac{\vec{u}^2}{c^2}}} \vec{u} \cdot \Delta \vec{x}_{AB} \right)^2 - \left( \Delta \vec{x}_{AB} + \frac{1}{\vec{u}^2} \left( \frac{1}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}} - 1 \right) \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right) \vec{u} + \frac{\vec{u}}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}} \Delta t_{AB} \right)^2 \\ = \frac{c^2}{1 - \frac{\vec{u}^2}{c^2}} \left(\Delta t_{AB} \right)^2 + \frac{2}{1 - \frac{\vec{u}^2}{c^2}} \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right) \left(\Delta t_{AB} \right) + \frac{1}{c^2 } \frac{1}{1 - \frac{\vec{u}^2}{c^2}} \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right)^2 \\ \quad \quad - \left( \Delta \vec{x}_{AB} \right)^2 - \frac{1}{\vec{u}^2} \left( \frac{1}{1 - \frac{\vec{u}^2}{c^2}} - \frac{2}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}} + 1 \right) \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right)^2 - \frac{\vec{u}^2}{1 - \frac{\vec{u}^2}{c^2}} \left( \Delta t_{AB} \right)^2 \\ \quad \quad - \frac{2}{\vec{u}^2} \left( \frac{1}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}} - 1 \right) \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right)^2 - \frac{2}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}} \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right) \Delta t_{AB} -2 \left( \frac{1}{1 - \frac{\vec{u}^2}{c^2}} - \frac{1}{\sqrt{1 - \frac{\vec{u}^2}{c^2}}} \right) \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right) \Delta t_{AB} \\ = \frac{c^2 - \vec{u}^2}{1 - \frac{\vec{u}^2}{c^2}} \left(\Delta t_{AB} \right)^2 - \left( \Delta \vec{x}_{AB} \right)^2 + \left( \frac{1}{c^2 - \vec{u}^2} + \frac{1}{\vec{u}^2} - \frac{c^2}{\left( \vec{u}^2 \right) \left( c^2 - \vec{u}^2 \right) } \right) \left( \vec{u} \cdot \Delta \vec{x}_{AB} \right)^2 \\ = c^2\left(\Delta t_{AB} \right)^2 - \left( \Delta \vec{x}_{AB} \right)^2 \)

 

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That is a valid point Harry. One that I have been wrestling with for many months now. Two people get in a ship and go to the Moon. One gets out on the Moon and the other goes back to Earth. The one on the Moon ages faster than the one on Earth. Isn't this, like your problem above, due to a change in their frame of reference?

 

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No, the person on the moon ages faster because the one on the moon is in a lower graviational field.

 

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Interesting!
This is what I would call a self fulfilling prophecy topped with a bizarre error.
If summarize what I read there I see first that because of SR an assumption is made that changing inertial frame versus not changing inertial frame is the reason why things are not symmetrical. Next it is shown that there is changing of inertial frame going on. And then that is used as evidence that things are not symmetrical. As I said, a self fulfilling prophecy because SR is not saying anything about that changing inertial frames has this effect. SR only says that laws of physics are identical in any given inertial system.
But then right at the end there is this:
“One of the consequences of the general theory is that clocks at high gravitational potential run more quickly than those at low potential. (So, for example, very accurate laboratory clocks on Earth run are observed to run faster when their altitude is increased.) In terms of Jane's local frame during the turn around, Joe is a long way overhead and so, according to her, his clocks run fast during that time, and he ages quickly. Further, Joe's 'height' above her depends on how far she has travelled, so his clocks run more quickly during the turn around in a long voyage. This is quite important, because proponents of the twin paradox sometimes argue that, whatever the effect of the turn around, it can be made negligible by making the journey far enough. Not so. The longer the journey, the greater the effect due to GR. (Similarly, in terms of the SR argument above, the longer the journey, the longer it takes for Jane's change of frames to be observed by Joe, and so the bigger effect.)”
This is just so unbelievably silly and bizarre reasoning. I cannot believe that anybody in their right mind could think this is in any way logical. I am not even going to dignify that with a response and forget this text is in there.
To go back to the self fulfilling prophecy: I strongly reject the notion that changing reference frames has anything to do with explaining away the twin paradox. I believe there is no paradox at all and my explanation makes a lot more sense.
Anybody else have some arguments for me to look at because I really like the way this is going.

 

Wow, go easy on me with the formulas please. To begin with I would need a circumscription of the symbols you are using because it has been 30 years since I last really used math. But I would prefer it if you used more English to try and convey your message to me for now. I do think however I could use somebody with your skills in the future, so let me know is you would be interested to help me then with developing some math for a couple of new ideas I have ?

 

I'm only an amateur but what I know is that the symmetry in the acceleration/deceleration in the travelling twins frame, dictates the non-symmetry nature of the twins paths, so there is no paradox.
That is the accepted explanation and makes plenty of sense.

 

http://en.wikipedia.org/wiki/Twin_paradox
Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason...".[1] Max von Laue argued in 1913 that since the traveling twin must be in two separateinertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se.[2] Explanations put forth by Albert Einstein and Max Borninvoked gravitational time dilation to explain the aging as a direct effect of acceleration.[3]

The twin paradox has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment.[A 1][A 2] It was also confirmed in particle accelerators by measuring time dilation of circulating particle beams.[A 3]

 

Again, from my limited understanding time dilation is caused by both speed and gravity...They can work together or against each other.
eg: GPS Satellites......
[1] Due to there speed, they appear heavier from an outside FoR. So time will appear to be dilated due to speed and gravity.
[2] Due to the fact that they are further from the CoG of the Earth, time will consequently go faster.


Let's convert that to the travelling twin...
[1] From the stay at home twin's PoV, his sibling is experiencing time dilation due to speed and being heavier and consequently more gravity.
[2] The travelling twin is also experiencing time going faster due to the near absence of the Earth's gravity.

Am I in error with any of the above scenarios?
I'm sure all effects are considered within the framework of SR/GR

Any errors, alterations or corrections in that layman's run down?

 

Yes, yes precisely. It has been experimentally verified that both(!) gravity and(!) velocity cause time dilation. The problem is only how to explain this effect. To find a reasonable explanation what is the underlying mechanism is that is causing time to slow down. The step from gravity causing time dilation to acceleration causing time dilation is small and generally accepted. But velocity ? That is a different beast. Velocity is R E L A T I V E. Without a reference frame where the velocity is relative to, velocity is 100% meaningless and it is impossible for velocity to have any effect, let alone time dilation. And if one assumes there is not ether then the only (and to me almost obvious) reference frame is a gravity field.

 

Don't think so because it has been experimentally verified that velocity also causes time to slow down besides acceleration/deceleration.

 

From an outside FoR, the faster we go, the more energy we obtain, the heavier we are, and the more gravity. E=Mc2

The following Q/A page has an excellent reply....
http://physics.stackexchange.com/qu...mb-rest-frame-special-where-does-it-come-from
The theory of special relativity is based on the principle that there are no preferred reference frames. In other words, the whole of Einstein's theory rests on the assumption that physics works the same irrespective of what speed and direction you have. So the fact that there is a frame of reference in which there is no motion through the CMB would appear to violate special relativity!

However, the crucial assumption of Einstein's theory is not that there are no special frames, but that there are no special frames where the laws of physics are different. There clearly is a frame where the CMB is at rest, and so this is, in some sense, the rest frame of the Universe. But for doing any physics experiment, any other frame is as good as this one. So the only difference is that in the CMB rest frame you measure no velocity with respect to the CMB photons, but that does not imply any fundamental difference in the laws of physics.

 

If I understand you correctly this is very close to what I am trying to say.
Two differences:
1: Gravity increase due to speed alone is not something I have considered yet but I assume that has already been accounted for with the Lorentz factor.
2: You say: "time going faster due to the near absence of the Earth's gravity". Yes(!) exactly and that then logically implies also: time going slower when gravity increases along the way.
In other words, this is my point or postulate:
Time dilation caused by velocity depends besides velocity also on the gravity field strengths experienced during travel.

 

Yes, yes, I understand an fully agree with that. I believe there is some miscommunication here. When I talk about a reference frame in the context of velocity induced time dilation I don’t mean a inertial reference frame in the context of SR. I mean reference frame literally as something where the speed is relative to, not necessarily an inertial reference frame. Maybe there is better word in English for this?

 

My point being that all this is taken into consideration anyway.
At least this is the way I have always interpreted it.

 

You seem to be laboring under a common misconception when it come to time dilation due to gravity. It is not related to any difference in gravity strength, but is instead related difference in height in the gravity field. To explain what this means, you can imagine a uniform gravity field that does not change in strength with height. If we place two clocks in this field at different heights, each clock experience the exact same gravity. However, the clock that is higher will still run faster compared to the lower clock.

Neither does acceleration cause time dilation as such. This is called the clock postulate. This has been tested on high speed centrifuges. It has been shown that the time dilation of radio-isotopes on such a centrifuge only depends on the speed at which the isotope is moving and is independent of the g-forces felt.

Any time dilation effects associated with acceleration follows the same rules as does gravity. For instance, if we put two clocks in the nose and tail of a rocket and then accelerate that rocket in such a way that both clocks experience exactly the same acceleration as measured by the clocks, then the clock in the nose of the rocket will run faster than the one in the tail, despite the identical acceleration felt by each. This is equivalent to the two clocks at different heights in a uniform gravity field.

 

Thanks Janus 58. (Correct as usual, although for greater clarity, I would say the clock with the higher gravitational potential, rather than height, runs faster.)

In post 12, I set up a "triplet paradox" with brothers B & C having exactly the same three acceleration wrt to stay at home brother A. (First acceleration was to get up to near speed of light, C, then coast. 2nd was to slow speed down and bring it for an instant to zero wrt to A and then get speed back towards A but when back near speed of light again, start second coast period. 3d was to end that 2nd coast period and come to rest again with A.)

Only difference was that brother C had coast periods of twice the duration as B's Thus B was Y year younger than A when standing together again and later when C reunited with A, he was about 2Y years younger. Again the ONLY difference between B & C's trips was the duration of their constant speed coast periods.

This, like Janus58 said*, shows it is speed and how long it lasts, not acceleration, that make their aging rates differ. That is also clear in the formulate for calculation to the time dilation ratio which is:

Δt' / Δt = 1 / sqrt[ 1 - {v^2 / C^2} ]

NOTE there is no acceleration or gravity terms in the equation ! but to get the Y years you need to integrate thru the speed varying (not coasting periods) effects too. Except for fact humans would be killed by huge accelerations the accelerations could by only a minute in duration and then compared to years of constant speed coasting any dilation of those three minutes can be neglected.

* "Neither does acceleration cause time dilation as such."

 

The point you're missing out on is that in order for the laws of physics to be identical in all inertial frames, including Maxwell's equations of electromagnetism, the distance and time separations between events must differ between observers in precisely such a way that changing reference frames does indeed result in asymmetries. If the Earth were accelerating on giant rockets instead of the astronaut, you'd get the opposite outcome.

 

One wild card variable, that traditional gravity brings, that does beyond GR, is pressure. The pressure created by the gravity of the sun, allows fusion in the core. The fusion is not space-time dependent, but pressure dependent. If it was space-time dependent, than motion via SR could also cause fusion.

The center of the earth is at about 7000C. Iron at that temperature should be a gas. But the pressure due to gravity causes it to become solid. If we traveled in a rocket, so the space-time contraction was the same as the core of the earth, iron at 7000C would not solidify. Space-time, alone is not enough to explain gravity.

If you look at the sun, as we go to the core; down the space-time well, as the pressure increases, the transitional frequencies of matter and energy increase all the way to gamma/fusion at the core. Notice material and energy time is getting faster, due to pressure; frequencies get faster, even though time is slowing via GR. The distances of interactions and wavelength contract in the same direction as space-time, but time/frequency goes the opposite way due to the impact of pressure.

Pressure is force/area with this force acting apart from space-time.

 

Did you deduce these claims from looking at the Einstein field equations, or did you just make them up? General Relativity both takes pressure into account when calculating spacetime curvature, and it explains why pressure builds up at the centers of planets and stars in the first place. Spacetime curvature does more than simply alter the trajectories of moving bodies, it also produces relative motion where none existed to begin with.

 

Interesting. Could you point me to a location where I can find info about these experiments? I have been looking, but can't find it.