# Special relativity and acceleration

=Al68;2302912] ]
The front and rear of an accelerating spaceship would be at two different (but not independent) velocities with respect to an inertial frame. This is a consequence of SR, not an assumption
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Subtle distinction there between different and independant

originally austin0
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In light of your interpretation I have a question.
What is the difference btween proper acceleration in the system and coordinate acceleration as viewed from another frame??

They are equal if the other frame is the instantaneously co-moving inertial frame. But since that frame is different from moment to moment, they are rarely used as reference frames in a scenario, except sometimes one of them. Proper acceleration depends on force applied (thrust) and the mass accelerated, and is the same value in any frame. Coordinate acceleration, like coordinate velocity, depends on reference frame
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I understand the definitions of proper and coordinate acceleration. That was not my question. If acceleration of a frame results in dilation what difference does the semantical distinctions make with regard to its observation from another inertial frame.
Ie: Whether you call it proper acceleration or coordinate acceleration or consider it a velocity differential due to acceleration,,, why is it not detectable in the acceleration testing performed so far???

original quote austin0
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I dont think this analogy really applies. It is not a question of 1 in =2.54 cm but 4.67cm in S being equivalent to 7.9cm in S'

But the reason for that is convention as well. It's based on the SR simultaneity convention. If the proper length of my spaceship is 20 ft long, and an observer in relative motion at 0.6c measures the spaceship length by using light signals, he would use the SR simultaneity convention to determine the location of each end, and determine the ends to be 16 ft apart at a specified moment. Assuming a constant invariant light speed, the length of the ship in the other frame is 16 ft by convention. Length contraction is a result of convention, and invariant light speed.Only in the same way as the standard method for converting inches to cm. The SR simultaneity convention, like other conventions, isn't a law of physics.
I think we completely disagree on this one. The Lorentz math is no more a convention than the inverse square proprtionality in gravity and electrostatics. There is no other possible expression. This is a description of fundamental aspects of reality and physicists of Andromeda would inevitably discover the same expressions although undoubtably with different conventional units.

These conventions are used in a physical theory not as claims, but as useful tools. Alternative conventions could be used, and the final results would be the same, if the theory using the other conventions was otherwise equivalent to SR/GR.
Thanks

Ich
Maybe I am once again misunderstanding but I thought that, as abstract constructions , the basis of both GR and electrodynamics was exactly that; assigning local values to points in the coordinate space?
GR is formulated via local quantities, the curvature. Those quantities cannot be measured strictly locally, by their very nature you need to survey some finite region to measure them.

Also with EM, there are local quantities that define the field at each point. But these are locally measurable.

And the point I'm trying to drive home is that all those locally measurable quantities are totally independent of these local GR quantities. Local curvature doesn't influence the local E-Field in the least, both are independent by their very mathematical definition - where vectors and tensors "live in the tangent space".
Does this mean the measured velocity ,both empirically and theoretically, is invariiant , in this context, the same up and down the potential gradient???
As long as they are measured in a local free falling frame: yes.

Al68
I understand the definitions of proper and coordinate acceleration. That was not my question. If acceleration of a frame results in dilation what difference does the semantical distinctions make with regard to its observation from another inertial frame.
Ie: Whether you call it proper acceleration or coordinate acceleration or consider it a velocity differential due to acceleration,,, why is it not detectable in the acceleration testing performed so far???
I'm not sure I understand what you mean by "not detectable", but the difference between proper acceleration and coordinate acceleration is not semantical. The coordinate acceleration of any object can be made equal to any arbitrary value simply by the choice of a suitable reference frame.
I think we completely disagree on this one. The Lorentz math is no more a convention than the inverse square proprtionality in gravity and electrostatics. There is no other possible expression. This is a description of fundamental aspects of reality and physicists of Andromeda would inevitably discover the same expressions although undoubtably with different conventional units.
Length contraction relies on not only the SR simultaneity convention, but the invariance of the speed of light, which is considered a law of physics. When I said that length contraction was based on convention, I should have said that the specific example (proper to coordinate length ratio) was based on convention, not the phenomena of length contraction itself. My bad.

Originally Posted by Austin0
I understand the definitions of proper and coordinate acceleration. That was not my question. If acceleration of a frame results in dilation what difference does the semantical distinctions make with regard to its observation from another inertial frame.
Ie: Whether you call it proper acceleration or coordinate acceleration or consider it a velocity differential due to acceleration,,, why is it not detectable in the acceleration testing performed so far???

=Al68;2307175]I'm not sure I understand what you mean by "not detectable", but the difference between proper acceleration and coordinate acceleration is not semantical. The coordinate acceleration of any object can be made equal to any arbitrary value simply by the choice of a suitable reference frame.
I will try to express myself more clearly. I was not suggesting that the difference between coordinate and proper acceleration was a matter of semantics.

You have a singular accelerating system and a unique set of observations as recorded in the system and an inertial frame. This is an unambiguous situation. Time dilation is either detected or it isnt as determined by comparison of proper elapsed time between the accelerated clocks and the inertial clocks after acceleration.
According to the actual real world tests performed so far to my knowledge,,,, there is no additional dilation due to acceleration. The dilation that has been recorded is totally accounted for by the dilation due to relative velocity. Ie: Due to the instantaneous velocities but with no additional factor due to the change in velocities.
This theorem posits that there is no dilation due to proper acceleration. There is no dilation due to coordinate acceleration. BUT there is additional dilation taking place due to the velocity differential resulting from acceleration.
That is what I meant about semantics. Simply calling it dilation due to a [Bvelocity[/B]differential from acceleration doesn't alter the fact that so far it is not detected.
I ,of course ,,cannot say that this is neccessarily true and is not simply a result of testing limitations etc etc. I also am taking the conclusions derived by the various mathematicians as valid without being able to perform the calculations myself. But can you simply dismiss the data in favor of a theoretical hypothesis???
Thanks

=Ich;2306276]GR is formulated via local quantities, the curvature. Those quantities cannot be measured strictly locally, by their very nature you need to survey some finite region to measure them.

OK this makes sense ,,,I wouldn't expect to be able to directly measure the field itself.

And the point I'm trying to drive home is that all those locally measurable quantities are totally independent of these local GR quantities. Local curvature doesn't influence the local E-Field in the least, both are independent by their very mathematical definition - where vectors and tensors "live in the tangent space".
Vectors and tensors may live in "tangent space" but clocks dont. SO I dont follow how these local measurements can be totally independant of the field or curvature either. I thought particles were totally effected by the curvature and local quantities both in there paths and periodicity.
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Previous austin0
Does this mean the measured velocity ,both empirically and theoretically, is invariiant , in this context, the same up and down the potential gradient???
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As long as they are measured in a local free falling frame: yes
Do you know of any actual tests done either from static locations at differing G altitudes or actual free fall tests????

Ich
I thought particles were totally effected by the curvature and local quantities both in there paths and periodicity.
See, that's why I keep reminding you not to think of broken clocks. A clock's "periodicity" is in no way effected bv local quantities. The best mathematical representation of "ticking rate" is indeed a vector (four-velocity), and as such lives in tangent space. Comfortably, I am told.

It's different with the path: local curvature defines how vectors, tensors and such change when being "transported" from one event to another. "Transport" can be a real transport, like moving a gyroscope around and observing how its axis - a vector - changes. Or a mathematical concept, like shifting A's four velocity through curved spacetime to the position of B's four velocity. That's the only way to compare vectors in GR, you have to bring both together, and it does matter how you do that. If the comparison shows that both don't point in the same direction: that's called time dilation.
Do you know of any actual tests done either from static locations at differing G altitudes or actual free fall tests????
I don't know any tests of light speed under awkward circumstances. But GPS, for example, depends on light behaving exactly like GR says, which, in turn, should coincide with what I say.

Al68
According to the actual real world tests performed so far to my knowledge,,,, there is no additional dilation due to acceleration. The dilation that has been recorded is totally accounted for by the dilation due to relative velocity. Ie: Due to the instantaneous velocities but with no additional factor due to the change in velocities.
This theorem posits that there is no dilation due to proper acceleration. There is no dilation due to coordinate acceleration. BUT there is additional dilation taking place due to the velocity differential resulting from acceleration.
That is what I meant about semantics. Simply calling it dilation due to a [Bvelocity[/B]differential from acceleration doesn't alter the fact that so far it is not detected.
I ,of course ,,cannot say that this is neccessarily true and is not simply a result of testing limitations etc etc. I also am taking the conclusions derived by the various mathematicians as valid without being able to perform the calculations myself. But can you simply dismiss the data in favor of a theoretical hypothesis???
I see your point here. And you're right. What you say is not only supported by the data, but by theory as well. That is the crux of the clock hypothesis.

But semantically, I would disagree with the way you word this: "There is no dilation due to coordinate acceleration", simply because, by definition, this is the equivalent of saying: "There is no dilation due to coordinate velocity changes", which you point out, is not true.

Gravitational time dilation is both predicted by theory and supported by data, there is no contradiction. But whether the detected dilation is "gravitational" or "velocity based" depends on which coordinate system we choose. It's the same effect either way, both in theory and practice. Using the phrase "gravitational time dilation" is just a convenient way to explain the dilation of clocks which are "stationary" in an accelerated frame, but is the same dilation that would be referred to as "velocity based" when observed from an inertial frame.

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austin0
I thought particles were totally effected by the curvature and local quantities both in there paths and periodicity.

=Ich;2309199]See, that's why I keep reminding you not to think of broken clocks. A clock's "periodicity" is in no way effected bv local quantities. The best mathematical representation of "ticking rate" is indeed a vector (four-velocity), and as such lives in tangent space. Comfortably, I am told.

It's different with the path: local curvature defines how vectors, tensors and such change when being "transported" from one event to another. "Transport" can be a real transport, like moving a gyroscope around and observing how its axis - a vector - changes. Or a mathematical concept, like shifting A's four velocity through curved spacetime to the position of B's four velocity. That's the only way to compare vectors in GR, you have to bring both together, and it does matter how you do that. If the comparison shows that both don't point in the same direction: that's called time dilation.
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See if I have this right. A clock at a specific location has a defined four-vector (periodicity). Translation to another location and comparison of four-vectors with a resident clock reveals that they dont point in the same direction =time dilation.

Is it not true that once colocated for comparison they will then have vectors that point in the same direction? That the time dilation is revealed by the difference in elapsed proper time , the clock reading. That the change in the moved clocks vector is not a SR result of its motion,, but is an incremental change due to the local curvature. Ie: measuring and responding to the local condition in the same way your gyroscope did.

I must be seriously misunderstanding here but I thought that tensors did not move but were constant values with fixed locations within the overall field??????

austin0_________________________________________________________________________
Do you know of any actual tests done either from static locations at differing G altitudes or actual free fall tests????
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I
don't know any tests of light speed under awkward circumstances. But GPS, for example, depends on light behaving exactly like GR says, which, in turn, should coincide with what I say
Good idea. But from what I have been able to find it was not explicit if the transit time compensation was calculated at exactly c or not. But it did seem to be a single figure which should mean that the time from earth to satellite is the same as satellite to earth.
Ill keep looking unless you have more defintie info.
Thanks

Ich
See if I have this right. A clock at a specific location has a defined four-vector (periodicity). Translation to another location and comparison of four-vectors with a resident clock reveals that they dont point in the same direction =time dilation.
That's what I wrote, but
1. my description is not entirely correct (in fact it's wrong, but ok for the purpose),
2. I meant "difference in clock rate" instead of "time dilation".
Is it not true that once colocated for comparison they will then have vectors that point in the same direction? That the time dilation is revealed by the difference in elapsed proper time , the clock reading. That the change in the moved clocks vector is not a SR result of its motion,, but is an incremental change due to the local curvature.
All true. But like motion, that is a change relative to some other observer. While there is no locally defined "absolute direction" of said vector, just like there is no absolute velocity.
I must be seriously misunderstanding here but I thought that tensors did not move but were constant values with fixed locations within the overall field?
If you take the spin direction of a gyroscope as an example: that can be thought as fixed to the gyroscope's position and moving with it. Generally, we're talking about http://en.wikipedia.org/wiki/Parallel_transport" [Broken] abstract mathematical procedure. Pure geometry, nothing to do with physics - except that SR and GR are also pure geometry, the latter at least with a coupling to mass.

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I have been following with great interest the discussions in this thread, the comments on the interaction of gravity with light being of particular interest. I wonder if the following possibility has ever been considered:

Suppose that the presence of a gravitational field resulted in a modification of the product of the electric permittivity ε and the magnetic permeability μ of free space, which is the quantity which determines the velocity of light in free space according to the relation c^2 = 1/( ε. μ ) .

Suppose further that this resulted in a refractive index in a gravitational field which was a function of that field, possibly even a linear function. That would mean that the deflection of a light beam could be attributed to refraction by a changing refractive index, rather than directly to a gravitational force.

All the relevant equations could be changed by the inclusion of suitable factors which linked the value of refractive index to the gravitational field, thereby modifying the value of c to produce the same end result in every case. One major difference would be that it would no longer be necessary to deal with the distortion of space-time, because the resulting deflections could then be attributed to effects caused by changes in the refractive index of free space due to gravitational fields.

This interpretation would not for example change the result of the Pound-Rebka Harvard Tower experiment. The time-dilation produced by the gravitational red-shift would instead be interpreted as a change in the value of c, caused by the change of the refractive index, (due to the gravitational field) between the source and emitter, leading to the mismatch between the photon energies.

Would the concept of ‘gravitational time dilation’ even be necessary if what was really happening were changes in the value of c caused by a variable refractive index. The term ‘gravitational red-shift’ however, would still be entirely appropriate, since the result would still ultimately be due to the gravitational field, but by modification of the refractive index rather than by ‘time-dilation’.

Ich
Would the concept of ‘gravitational time dilation’ even be necessary if what was really happening were changes in the value of c caused by a variable refractive index.
Of course. The idea of an effective refraction index is not new, a quick search gave me http://homepages.ecs.vuw.ac.nz/~visser/Seminars/Conferences/refractive-index-2.pdf" [Broken].
GR can't be reduced to a scalar theory, as the refractive index would be. So that's not an intrpretation, it's a new theory which contradicts some experiments (I don't know which at the moment, you can look it up in MTW's Gravitation).
Even in cases where a scalar quantitiy is enough, it is not evident how a refractive index should produce real time dilatation, the one accumulating over measurement time.

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Thanks for the quick reply and the reference. My comment on the 'gravitational time dilation' was only in the context of the Harvard Tower experiment. Just to clarify the gist of my argument:

Suppose that a rocket containing an atomic clock was accelerated on an outward journey and was then decelerated to return to the starting point. If the refractive index was affected as suggested by the gravitational fields generated in the acceleration/deceleration periods, then the clock frequency would be reduced for both those periods and would of course show a corresponding cumulative time dilation.

Ich
Suppose that a rocket containing an atomic clock was accelerated on an outward journey and was then decelerated to return to the starting point. If the refractive index was affected as suggested by the gravitational fields generated in the acceleration/deceleration periods, then the clock frequency would be reduced for both those periods and would of course show a corresponding cumulative time dilation
But time dilatation is proportional rather to the total time you spend on the journey, not to the acceleration phases. You can't model this "SR" time dilatation with a local refraction index.
You can express time dilatation in static spacetimes with a single parameter, but I don't see how this would be due to a different refraction index. But as I said, the theory can't replace GR anyway.

It does: and this is what I describe above in the previous post.

Note that you cannot simply compare an accelerated observer with an unaccelerated observer, because in that case you also get an increasing velocity difference, and that dominates any time dilation.

However, if you have a long spaceship experiencing a constant acceleration at all points, it turns out that the front has slightly less acceleration than the back, and there is a time dilation between the front and the back of the ship... but no change in the distance between them, as measured by anyone on the ship. THIS is what turns out to be exactly analogous to the time difference of two observers at different altitudes in a gravitational field.

Cheers -- sylas
Hi sylas I am still thinking over this topic.
I did a work up of a hypothetical case but my math is rusty so I thought I would run it by you.
Inertial frame F
Accelerating System S'
rest L'= 1 km
a= 1000g= 10km /s$$^{2}$$

Range .6c ===> .7c
.7c-.6c =.1c = 3 x 10$$^{4}$$km/s

Time dt= (3 x 10$$^{4}$$km/s)/(10km/s) =3000 s

Contraction v$$_{i}$$=.6c ------- $$\gamma$$=1.25 --- = L'$$_{0}$$=.8 km
v$$_{f}$$ =.7 -------- $$\gamma$$= 1.4 --- =L'$$_{1}$$ =.71km

Difference in length over course of acceleration = .09 km
.09km/ 3000s = 3 x 10 $$^{-5}$$ km /s

relative velocity between front and back v$$_{fb}$$= (3 x 10 $$^{-5}$$ km /s) /(3 x 10 $$^{5}$$km /s ) = 10$$^{-10}$$c
Additive average relative velocity between front and back = (.65+10$$^{-10}$$)+ .65c = .65+ (1.7316 e$$^{-10}$$ )

average velocity difference v$$_{d}$$= 1.7316 e$$^{-10}$$ c
avg $$\gamma$$= 1 +( 2.9484 x 10 $$^{-20}$$ ) between front and back

Relative to inertial frame F ,,, S' avg v=.65c $$\gamma$$= 1.32
dt/1.32 = 3000/1.32 = 2,273 s = overall elapsed time on S'

2,273 / 1 +( 2.9484 x 10 $$^{-20}$$ ) = 6.782 x 10 $$^{-17}$$ s
elapsed time difference between back and front.

As I said I am rusty and could have easily dropped an exponent or counted all the zeros or 9's on the calculator screen wrong but does this seem in the ballpark???
Or is there some other fundamentally different way to calculate?
I assumed constant acceleration as observed in the inertial frame , of course the calculated (a) factor wouldn't neccessarily be healthy for humans but it made for smaller numbers
Thanks cia0

Silas:
I was notified that your reply to AustinO (the quote in message #64) pertained to a question that I raised in message #63 and indeed it does. I get the point about the acceleration on a long rocket being different at each end, because of time dilation over the length of the thing, due to the gravitational field which it generates being different from end to end.

Can you give me some idea of what the polar diagram for the field looks like? Surely it cannot have anything to do with the position of the propulsive force. Suppose for example the source of this force was situated at the centre of the rocket, rather than at the rear end, would that change the polar diagram of the gravitational field? I can't imagine that it would.

For example what would the field look like if the accelerating object was not a rocket but a spherical mass with the propulsive force being a point source at the centre of the sphere? Don't ask me how such an arrangement could be devised without a radial hole to the centre for the gas or the ions or whatever to escape and provide thrust, but let's assume that it is in the realms of mythical frictionless surfaces and perfectly reflecting ones along with the other mythical assumptions which are made to illustrate a basic principle.