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Thread: Pseudogravitational field in General Relativity

  1. #1 Pseudogravitational field in General Relativity 
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    Ok, as I understand it, we can take the equation for gravitational time dilation (in terms of coordinate time) in GR:

    Gravitational time dilation - Wikipedia, the free encyclopedia

    Then, for the pseudogravitational field of an object accelerating by non-gravitational means, we can substitute

    (r times acceleration) for because acceleration in a gravitational field is equal to , which is / r

    So the equation becomes
    And that is what we use to calculate time dilation for an acceleration very far away from us that is not due to gravitation, but just an ordinary acceleration. Am I right about that so far, or am I missing something?


    Now my confusion is this: for sufficiently long distances (by which I mean large "r" values), wouldn't that equation be capable of yielding imaginary values (the square root of a negative number)?

    I am trying to figure out the math behind the twin paradox solution on this website speedfreak linked to on another thread. The explanation makes a lot of sense, but I want to work out the math for it so I can really understand what is happening.

    The Twin Paradox: The Equivalence Principle Analysis






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  3. #2  
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    I am confused!
    Quote Originally Posted by kojax View Post
    Ok, as I understand it, we can take the equation for gravitational time dilation (in terms of coordinate time) in GR:
    But let me first give a gentle reprimand; an equation MUST include and "equals" operator, so it it not at all clear what you mean by "gravitational time dilation in your statement. What you have is an expression, not an equation

    an object accelerating by non-gravitational means, we can substitute

    (r times acceleration) for

    So the equation becomes
    Again, this is not an equation. But here is one - you seem to be suggesting (correct me if I am wrong) that which (multiplying through by ) implies that which (multiplying through by ) implies that .

    Although I am no physicist, this seems extremely unlikely to me


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  4. #3  
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    Quote Originally Posted by kojax View Post
    Ok, as I understand it, we can take the equation for gravitational time dilation (in terms of coordinate time) in GR:

    Gravitational time dilation - Wikipedia, the free encyclopedia

    Then, for the pseudogravitational field of an object accelerating by non-gravitational means, we can substitute

    (r times acceleration) for because acceleration in a gravitational field is equal to , which is / r

    So the equation becomes
    And that is what we use to calculate time dilation for an acceleration very far away from us that is not due to gravitation, but just an ordinary acceleration. Am I right about that so far, or am I missing something?

    You are missing something. [quote] In the gravitational time dilation formula, r represents the distance from the center of the mass M, and the resulting time dilation is as measured by an observer far removed from the gravity of M.

    In


    a stands for the acceleration of gravity at distance r from the center of mass and really should be labeled

    It is dependent on r and is not the same thing as the acceleration of an object not due to gravity.

    So really, your formula:



    should actually be



    Where ag decreases by the square of r.

    All you did was restate the gravitational time dilation equation in a different form.

    In addition, you can't just switch the meaning of r from the distance from the center of M, to the distance of the accelerating object from the observer.
    Now my confusion is this: for sufficiently long distances (by which I mean large "r" values), wouldn't that equation be capable of yielding imaginary values (the square root of a negative number)?




    That being said. there is something known as the Rindler horizon for accelerating objects. It acts something like an event horizon, where anything greater than a certain distance from the accelerating observer that is in the direction opposite of his acceleration is hidden from him and he can know nothing about it.
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  5. #4  
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    Acceleration itself leads to no time dilation or length compression, according to the muon storage experiments.

    From the wikipedia article Time dilation of moving particles - Wikipedia, the free encyclopedia:

    Bailey et al. (1977) measured the lifetime of positive and negative muons in the CERN Muon storage ring. Time dilation was measured by sending muons around a loop. ... The so-called clock hypothesis of relativity, according to which the extent of acceleration doesn't influence the value of time dilation, was also confirmed in this experiment, as well as in the one of Roos et al. (1980).
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  6. #5  
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    Quote Originally Posted by Janus View Post


    In


    a stands for the acceleration of gravity at distance r from the center of mass and really should be labeled

    It is dependent on r and is not the same thing as the acceleration of an object not due to gravity.

    So really, your formula:



    should actually be



    Where ag decreases by the square of r.

    All you did was restate the gravitational time dilation equation in a different form.

    In addition, you can't just switch the meaning of r from the distance from the center of M, to the distance of the accelerating object from the observer.
    Now my confusion is this: for sufficiently long distances (by which I mean large "r" values), wouldn't that equation be capable of yielding imaginary values (the square root of a negative number)?


    Ok. Yeah. I see what you're saying. If I'm further from the center of a large object with strong gravity than another person is, that doesn't necessarily guarantee that the other person's distance from me is equal to the difference between our radius's. Maybe they're on the other side of the object and closer to it.

    However, there is still a distance aspect associated with psuedo-fields, right? Clearly it is not the one I tried to use, so I'll have to keep looking. As I mentioned, mostly I just want to figure out the maths behind the twin paradox. I think I would find the explanations more satisfying if I could calculate them (or whatever is possible. I know sometimes GR gets too complicated to just enter into a calculator.)



    That being said. there is something known as the Rindler horizon for accelerating objects. It acts something like an event horizon, where anything greater than a certain distance from the accelerating observer that is in the direction opposite of his acceleration is hidden from him and he can know nothing about it.
    This is good to know about. I'm trying to google it to find more information. Wiki only discusses it with respect to black holes, which I can't really use to understand non-gravitational acceleration.

    Quote Originally Posted by Guitarist View Post
    I am confused!
    Quote Originally Posted by kojax View Post
    Ok, as I understand it, we can take the equation for gravitational time dilation (in terms of coordinate time) in GR:
    But let me first give a gentle reprimand; an equation MUST include and "equals" operator, so it it not at all clear what you mean by "gravitational time dilation in your statement. What you have is an expression, not an equation
    Yeah. Sorry about that. I didn't know how to do subscripts in tex, so I was worried I wouldn't be able to post it right if I tried to write it out completely, but Janus' post conveniently contained a subscript, so I now I can see how to do it from his example.

    The equation I'm after is



    an object accelerating by non-gravitational means, we can substitute

    (r times acceleration) for

    So the equation becomes
    Again, this is not an equation. But here is one - you seem to be suggesting (correct me if I am wrong) that which (multiplying through by ) implies that which (multiplying through by ) implies that .

    Although I am no physicist, this seems extremely unlikely to me
    Taking this, , and dividing both sides by we would get , which is the standard equation for Newtonian gravity. Admittedly I'm wrong interpreting distance from the observer as being r, though.

    I'll be careful to write that part as now that I know how to make subscripts. Tex is amazingly useful on these forums.
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  7. #6  
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    Ok, I think I'm starting to see how this works.

    is the formula for potential energy due to gravity. Or more generally/correctly, the equation is

    For simple calculations on Earth for, like engineering stuff, we approximate the field to be uniform and use . So since a pseudo gravitational field is uniform, I'm thinking we can just substitute the second formula for the first. We just remove the mass of the "falling" object, change height to distance, and use the rest, substituting instead of

    So And it turns out that's not any different from what I had, but at least I'm deriving it more properly.

    Anyway, guitarist's thread on mass-energy equivalence got me thinking about this.

    Mass-Energy equivalence in SR

    Einstein's argument that an object's total energy changes when it becomes time dilated, and that it could be related to the equation for kinetic energy if we set , implies to me that the amount of time dilation something experiences due to motion should always be related to the effect that motion has on its energy.
    Last edited by kojax; May 3rd, 2012 at 09:50 PM. Reason: edit, change it to be [tex]t_0=t_f\sqrt{1-\frac{2a_gd}{c^2}}[/tex] instead of [tex]t_0=t_f\sqrt{\frac{2a_gd}{c^2}}[/tex]
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  8. #7  
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    Okay.

    If you are an accelerating clock, then the time dilation you measure in another clock separated by a distance along the direction of acceleration is



    where to is the rate you see the clock running, tc the clock's proper rate, a is the acceleration you are under, and d the distance between you (positive if the clock is in the direction of the acceleration and negative if it is in the opposite direction).

    So for example, if you are in a 1 light-second long rocket that is accelerating at 1g, a clock at the nose of the rocket runs 1.000000033 times faster than an identical one in the tail.

    However, I don't see this helping much. Because this assumes that the other clock is not moving with respect to you as measured in your frame. If you want to figure out the time dilation of a clock you are accelerating towards or away from, then you also have to take the time dilation due to your relative velocity into account.

    Even then, you only get the the total time dilation at that instant. If you want to know the difference in time accumulation over a period of time, you have to account for the fact that not only does the relative speed between you change over the period, but so does the distance between you. This makes the calculation much more complicated.
    kojax likes this.
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  9. #8  
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    Are the other effects just added together with the distance effect, or is it more complicated than that? In the twin paradox, it's possible to set up the thought experiment so that distance would appear to be the dominant factor. Suppose we just call them twin 1 and twin 2, and twin 1 is the one making the trip away and back to twin 2's home world. If twin 1 travels for 6 years (according to twin 1's perspective, longer for twin 2) at a relativistic speed like .8c and then turns around over the course of 5 days, the time dilation effects associated with 5 days at speeds smaller than 0.8c could pretty much be ignored since they're so small compared to the 6 years (12 years round trip) spent going at full speed. Also the distance twin 1 traverses over 5 days of deceleration/acceleration would be trivial.

    Well, except for the change in perception of distance due to Lorentz contraction as twin 1's velocity changes relative to twin 2. Does that count as motion?
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    Ok, using my unit of distance as the "light year", and my unit of speed as "light year per year", twin 1 has been traveling at .8 light years per year for 6 years. Total distance of 4.8 light years. If she slows down and then speeds up again in 5 days, then her rate of deceleration is 0.32 light years per year per day (to go from 0.8 c away to 0.8 c toward), or 365 * 0.32 = 116.8 light years per year per year. C = 1 light year per year.

    So using these units

    a = 116.8 light years per year per year
    d = 4.8 light years
    c = 1 light year per year

    entering those values into we get or or

    For 5 days, twin 2 appeared to age at 561.64 times his normal rate, aging 2808.2 days or 7.69 years. That leaves out the additional effects from instantaneous velocity, etc.

    Length contraction at .8c is 0.6, so in terms of ordinary special relativity, twin 2 should think that twin 1 was traveling 8 light years distance each way instead of 4.8 light years each way, making it a total of 10 years each way, or 20 years round trip . So twin 2 thinks the round trip took 20 years. Twin 1 thinks the round trip took 12 years. And we've got 7.69 years worth of aging happening over a span of 5 days. It looks like this formula comes pretty close to solving the puzzle. If we used my last formula, the result would have been an imaginary number, so this is clearly better.
    Last edited by kojax; May 4th, 2012 at 12:03 PM. Reason: added unit titles
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    Quote Originally Posted by kojax View Post
    Now my confusion is this: for sufficiently long distances (by which I mean large "r" values), wouldn't that equation be capable of yielding imaginary values (the square root of a negative number)?
    No, because as r becomes very large, the fraction as a whole becomes very small. In other words, the further away you are from a gravitational source, the smaller the time dilation effect will be, just as one would expect.
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    Quote Originally Posted by Janus View Post
    Okay.

    If you are an accelerating clock, then the time dilation you measure in another clock separated by a distance along the direction of acceleration is



    where to is the rate you see the clock running, tc the clock's proper rate, a is the acceleration you are under, and d the distance between you (positive if the clock is in the direction of the acceleration and negative if it is in the opposite direction).
    Is there also a time dilation effect associated with other objects accelerating relative to the observer, or is it only present when the observer them self is accelerating?
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  13. #12  
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    Quote Originally Posted by kojax View Post
    [

    Is there also a time dilation effect associated with other objects accelerating relative to the observer, or is it only present when the observer them self is accelerating?
    No. This is known as the clock postulate. This has been tested by placing radioisotopes in a high-speed centrifuge. The rate at which they decay only depended at the speed they spun and was not affected by the acceleration they were under. (By adjusting the speed and arm length of the centrifuge, you can create different combinations of acceleration and speed. For example, you can have different speeds with the same acceleration or different accelerations with the same speed.)
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    You know, the funny thing about that test, though.... if you think about it.... is that it would yield the same result even if you were an observer being spun around by the centrifuge, because you would always spend equal amounts of time accelerating toward and away from your target. At least if the target clock you were attempting to observe were located on the same plane as the circular path you were traveling.

    The only difference is that you would be physically slightly further away when you were accelerating toward, due to being physically located at the back of the ring, and slightly closer when accelerating away, due to being located toward the front of the ring.

    Suppose the effect were strong enough that it added/subtracted .2 (by which I mean, suppose ), so for x amount of time you see the clock running at 1.2 times its normal speed, and then for the same amount of time you see it running at 0.8 times its normal speed. 1.2x + .8x = 2x Just as expected.

    On the other hand, I was googling the "clock postulate" and came across this site that mentions the test, and states that they measured the effect up to g's,.. so I guess even thet very small amount of difference created by the differing distances would be detectable at such a high speed.

    Does a clock's acceleration affect its timing rate?

    Although the clock postulate is just that, a postulate, it has been verified experimentally up to extraordinarily high accelerations, as much as 1018 g in fact (see the faq What is the experimental basis of Special Relativity?). Of course, the postulate also speaks of more than acceleration, it speaks of all derivatives of v with respect to time. But still it can be shown to be a reasonable thing to assume, because it leads to something that has been tested in other ways, as we'll see in the next section.
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    Quote Originally Posted by kojax View Post
    ...Now my confusion is this: for sufficiently long distances (by which I mean large "r" values), wouldn't that equation be capable of yielding imaginary values (the square root of a negative number)? ...
    If it helps see the
    Simple inference of time dilation due to relative velocity on wikipedia. The related expression comes from Pythagoras' theorem. The base of a right-angled triangle is your speed v as a fraction of c, the hypotenuse is the light path where c=1 in natural units, and the height is the Lorentz factor, with a reciprocal distinguishing length contraction and time dilation. Increase v and the hypotenuse goes flatter, but it's "restricted" in that it can't go flatter than horizontal.
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    Quote Originally Posted by kojax View Post
    You know, the funny thing about that test, though.... if you think about it.... is that it would yield the same result even if you were an observer being spun around by the centrifuge, because you would always spend equal amounts of time accelerating toward and away from your target. At least if the target clock you were attempting to observe were located on the same plane as the circular path you were traveling.

    The only difference is that you would be physically slightly further away when you were accelerating toward, due to being physically located at the back of the ring, and slightly closer when accelerating away, due to being located toward the front of the ring.

    Suppose the effect were strong enough that it added/subtracted .2 (by which I mean, suppose ), so for x amount of time you see the clock running at 1.2 times its normal speed, and then for the same amount of time you see it running at 0.8 times its normal speed. 1.2x + .8x = 2x Just as expected.

    On the other hand, I was googling the "clock postulate" and came across this site that mentions the test, and states that they measured the effect up to g's,.. so I guess even thet very small amount of difference created by the differing distances would be detectable at such a high speed.

    Does a clock's acceleration affect its timing rate?

    Although the clock postulate is just that, a postulate, it has been verified experimentally up to extraordinarily high accelerations, as much as 1018 g in fact (see the faq What is the experimental basis of Special Relativity?). Of course, the postulate also speaks of more than acceleration, it speaks of all derivatives of v with respect to time. But still it can be shown to be a reasonable thing to assume, because it leads to something that has been tested in other ways, as we'll see in the next section.

    The "target" is the axis of the centrifuge, and the test sample is always accelerating towards it.
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  17. #16  
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    How does this relate to GPS satellites losing time? Is a gravitational orbit different from a centrifuge in GR? I have been trying to understand better how gravity differs from other forms of acceleration.

    I'm still trying to wrap my head around this. It's weird to think that dime dilation due to acceleration would be a one-way street like that. It helps for me to apply the Twin Paradox to everything, so I can see that gravitational time dilation making the clock in the center of the centrifuge tick faster for the accelerating particles is how the differing perspectives are resolved. Still weird, but at least relativity continues to make more sense the more I investigate it. That's reassuring.
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