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Thread: Gravitational Time Dilation

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    Gravitational Time Dilation

    In developing GR, Einstein came up with the principle of equivalence which relates the acceleration of gravity with the acceleration of an observer in flat spacetime, from which came the Schwarzschild metric. At this time I am not contending the equivalence principle or the form of the metric, but only that of the variable used in the metric 1 - r_s / r = 1 - 2 G M / (r c^2). It can be seen in this derivation of the Schwarzschild metric, toward the bottom of the Wiki page, that that particular variable itself is not so much derived as it is inferred only by the weak field approximation to find K and S. But while the weak field approximation does give 1 - 2 G M / (r c^2) in general, it could also just as easily be 1 / (1 + 2 G M / (r c^2)) or sqrt(1 - 4 G M / (r c^2)), just as a couple of quick examples, so we will set about here to determine it precisely.

    Let's consider the gravitational time dilation of a hovering observer at a radius r in a gravitational field, which we will designate z_r. If light is emitted by a distant observer to that point in the field, successive pulses will be emitted by the distant observer, each travelling radially to the hovering observer at r. Since each pulse travels an identical path to the hovering observer, taking the same amount of time, then according to the distant observer, each will be received by the hovering observer at the same rate at which they were originally emitted, so at the same frequency from the distant observer's point of view. But because the hovering observer's clock is gravitationally time dilated, the hovering observer will observe a greater frequency of f_observed / f_emitted = 1 / z_r.

    Any frequency emitted by the distant observer will be increased according to the hovering observer by 1 / z_r, so the energy of the light has increased by this proportion between the local measured values by each of the observers. GR associates this increase in energy to that of the local time dilation observed of a freefaller falling from rest at infinity with 1 / z_r = gamma_kinetic, whereby z_r = sqrt(1 - (v/c)^2), v being the locally measured speed. From this GR infers the weak field approximation where v^2 = 2 G M / r, gaining z_r = sqrt(1 - (v/c)^2) = sqrt(1 - 2 G M / (r c^2)). From this we find that at the radius r_s = 2 G M / (r c^2), the freefaller will achieve light speed locally. Since nothing can travel faster than light speed, then if the process were reversed, if an object or even light were emitted from r_s or below, it will cannot escape, so r_s is an event horizon.

    But there is a problem with this. v^2 = 2 G M / r is only the weak field approximation for large r, and it is being applied at all r, the same for the locally measured speed as with Newtonian gravity. For instance, if a freefaller were to start falling with greater than zero initial speed, then they would achieve light speed sooner, before reaching r_s. But locally the field is SR, and no hovering observer can ever measure the speed of a freefaller to be greater than c. For this and other reasons, z_r = sqrt(1 - 2 G M / (r c^2)) cannot be the exact relation. So now we will examine further what the relation should be.

    Let's say that a clock is travelling past a hovering observer at r with a locally measured speed v. Since SR is valid locally, the hovering observer measures a time dilation of the clock of z_kinetic = sqrt(1 - (v_loc/c)^2). A distant observer also measures a gravitational time dilation of the hovering observer of z_r. Since there is no simultaneity difference between the distant and hovering observer, the distant observer, then, will observe a time dilation of the clock equal to that of the hovering observer, but that is also lessened by the gravitational time dilation between them, or z_clock = z_kinetic * z_r, directly multiplying the two time dilations to find what the distant observer measures of the clock.

    Similarly, let's say that at the radius r in the field, there exists another small gravitating mass far from the hovering observer. According to the hovering observer, then, another hovering observer that is a distance d from the second mass will have a time dilation of z_d, otherwise both observing no difference in time dilation from the first mass because they are at the same radius from that mass. To the distant observer, then, the time dilation observed of the second hovering observer will also be z_d, but lessened by the time dilation of the first mass as well, so z_d * z_r. Likewise, if we have say 4 such masses labelled a, b, c, and d, then the total time dilation the distant observer measures of a hovering observer will be z_a * z_b * z_c * z_d, depending upon where the hovering observer is positioned to each of those masses.

    Okay, so now let's say we have a point mass with mass m and a hovering observer at distance r from that mass. The distant observer will measure z_{m,r} for the time dilation of the hovering observer. Now let's place an identical mass directly next to the first. The total mass is now 2 m, so the distant observer will now measure a time dilation of the hovering observer of z_{2 m, r} = z_{m,r} * z_{m,r}. These two relations must be mathematically equivalent in terms of their masses, and as far as I can tell, there is one and only one way they can be, and that is with the mathematical relation j^(2 m k) = j^(m k) * j^(m k), where j is some constant and k depends upon r, but in this case r is the same for both. Since it doesn't matter what j is, as long as it is non-zero and non-unity and thereby only changing the value of k in accordance, we will make it (1/e)^(2 m k) = (1/e)^(m k) * (1/e)^(m k). In terms of the time dilation in the weak field, then, we get z_r = 1 / e^(m k), where z_r = sqrt(1 - 2 G M / (r c^2)) according to GR, so must approximate z_r = 1 - G M / (r c^2), only 1 / e^(G M / (r c^2)) will give this value, making k = G / (r c^2). There are no further factors of M that can be added to this in any way without interupting the direct relation between the masses we had before, leaving only the possibility of further factors in relation to k in terms of r and some other constant that might be added, but this is highly unlikely, so this relation should be exact as it stands.

    This now makes the variable in the Schwarzschild metric [1 / e^(G M / (r c^2))]^2 = 1 / e^(2 G M / (r c^2)) rather than (1 - 2 G M / (r c^2)). One immediate consequence we can see come of this new relation is that with 1 - 2 G M / (r c^2), the time dilation falls to zero for a hovering observer at r_s = 2 G M / c^2, while with 1 / e^(2G M / (r c^2)), it falls to zero only at r = 0, so there is no event horizon and no singularity, unless a mass is capable of collapsing completely to a point. There is still a redshift of light emitted from a body, more extreme depending upon where it originates in the field, but it can always escape, and there is no reversing of space and time at any place in the field as would occur below an event horizon.

    Now, if the time dilation of a hovering observer at r is determined by a point mass according to a distant observer with z_r = 1 / e^(G M / (r c^2)), then what about a uniform density sphere? The total time dilation of the hovering observer will be determined in the same way as found before by simply multiplying the time dilations of each of the individual point masses,

    z_total

    = z_{m,r1} * z_{m, r2} * z_{m, r3} * ...

    = 1 / e^(G m / (r1 c^2)) * 1 / e^(G m / (r2 c^2)) * 1 / e^(G m / (r3 c^2)) * ...

    = 1 / e^[(G m / (r1 c^2)) + (G m / (r2 c^2)) + (G m / (r3 c^2)) + ...]

    But the part in the brackets is just the same as for integrating the energy for Newtonian gravity, which is

    G m / r1 + G m / r2 + G m / r3 + ... for each point mass within a uniformly dense sphere

    = G M / r where r is the distance from the center of the total mass M, giving

    z_total = 1 / e^(G M / (r c^2))

    the same as that for a point mass.

  2. #2
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    Quote Originally Posted by grav View Post
    In developing GR, Einstein came up with the principle of equivalence which relates the acceleration of gravity with the acceleration of an observer in flat spacetime, from which came the Schwarzschild metric. At this time I am not contending the equivalence principle or the form of the metric, but only that of the variable used in the metric 1 - r_s / r = 1 - 2 G M / (r c^2). It can be seen in this derivation of the Schwarzschild metric, toward the bottom of the Wiki page, that that particular variable itself is not so much derived as it is inferred only by the weak field approximation to find K and S. But while the weak field approximation does give 1 - 2 G M / (r c^2) in general, it could also just as easily be 1 / (1 + 2 G M / (r c^2)) or sqrt(1 - 4 G M / (r c^2)), just as a couple of quick examples, so we will set about here to determine it precisely.
    Actually, they show that the form must be K(1+1/Sr), for some constants K and S. Solving for these constants gives the standard form.

    To first order, you are correct. (1-2GM/arc^2)^(a) will give the correct Newtonian limit for any a. If we are strictly talking about time dilation, this corresponds to the PPN parameter gamma.

    I personally like the idea of a=4, but this radically breaks the other parameters of the PPN formalism if no other effects are considered.

    http://relativity.livingreviews.org/.../fulltext.html

    But there is a problem with this. v^2 = 2 G M / r is only the weak field approximation for large r, and it is being applied at all r, the same for the locally measured speed as with Newtonian gravity. For instance, if a freefaller were to start falling with greater than zero initial speed, then they would achieve light speed sooner, before reaching r_s. But locally the field is SR, and no hovering observer can ever measure the speed of a freefaller to be greater than c. For this and other reasons, z_r = sqrt(1 - 2 G M / (r c^2)) cannot be the exact relation. So now we will examine further what the relation should be.
    This problem only arises is you apply this incorrect interpretation of v^2=2GM/r. The problem is actually from ignoring the impact of special relativity. We need to convert from velocities to rapidity, and then we will find that the rapidity increases at the same rate regardless of the actual velocity. (assuming radial motion) Then the rapidity increases without bound at the same horizon location, only actually reaching c at the horizon regardless of the initial velocity.

    In the slow limit rapidity is equal to velocity.

    The remainder of your post appears to depend on improperly treating the difference when going from one to the other.
    Last edited by utesfan100; 2012-Apr-10 at 04:11 PM. Reason: Fix a in first order estimate, and include link for PPN formalism.

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    Quote Originally Posted by utesfan100 View Post
    Actually, they show that the form must be K(1+1/Sr), for some constants K and S. Solving for these constants gives the standard form.

    To first order, you are correct. (1-2GM/arc^2)^(a) will give the correct Newtonian limit for any a. If we are strictly talking about time dilation, this corresponds to the PPN parameter gamma.

    I personally like the idea of a=4, but this radically breaks the other parameters of the PPN formalism if no other effects are considered.

    http://relativity.livingreviews.org/.../fulltext.html



    This problem only arises is you apply this incorrect interpretation of v^2=2GM/r. The problem is actually from ignoring the impact of special relativity. We need to convert from velocities to rapidity, and then we will find that the rapidity increases at the same rate regardless of the actual velocity. (assuming radial motion) Then the rapidity increases without bound at the same horizon location, only actually reaching c at the horizon regardless of the initial velocity.

    In the slow limit rapidity is equal to velocity.

    The remainder of your post appears to depend on improperly treating the difference when going from one to the other.
    Thanks, but I'm not sure what you are saying about rapidity. What I am presenting actually does not refer to rapidity or speeds at all for that matter, but determines gravitational time dilation in respect to stationary point masses at different locations to an observer as z_ total = z_{m_1, r_1} * z_{m_2, r_2} * z_{m_3, r_3} * ... , which as derived from the example I gave and considering the low order approximation, works out to z_{M, r} = 1 / e^(G M / (r c^2)) for each point mass and the same for a spherical mass of uniform density.

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    The Schwarzschild metric applies only to the outer region of the mass of spherical symmetry.

    There is: B (r) = 1 / A (r) = 1 - a/r, exactly.
    Constant 'a' is determined from Newton.

    Sometimes people use the functions of type exp( m(r) ) instead of A(r) and B(r), and then it is clear that they can not be negative:
    exp(-m) = 1 - a/r, => r > a = 2m.

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    Quote Originally Posted by grav View Post
    In developing GR, Einstein came up with the principle of equivalence which relates the acceleration of gravity with the acceleration of an observer in flat spacetime, from which came the Schwarzschild metric. At this time I am not contending the equivalence principle or the form of the metric, but only that of the variable used in the metric 1 - r_s / r = 1 - 2 G M / (r c^2). It can be seen in this derivation of the Schwarzschild metric, toward the bottom of the Wiki page, that that particular variable itself is not so much derived as it is inferred only by the weak field approximation to find K and S.
    But and are constants, so they are the same everywhere. The use of the weak field approximation is only for finding the value of these constants.

    In a nutshell the thinking goes like this: We derive the schwarzschild metric but are short the values of two constants that we cannot determine. But if we go far from the central mass, we know that it must approach the newtonian solution, so we use that solution there to find the values of those constants. Since they are constant, they have the same value everywhere, so we have solved the schwarzschild metric everywhere (they also have the same value right next to the black hole, we just don't have a direct way of determining them there, so we determine their values very far from the black hole).

    But there is a problem with this. v^2 = 2 G M / r is only the weak field approximation for large r, and it is being applied at all r, the same for the locally measured speed as with Newtonian gravity. For instance, if a freefaller were to start falling with greater than zero initial speed, then they would achieve light speed sooner, before reaching r_s. But locally the field is SR, and no hovering observer can ever measure the speed of a freefaller to be greater than c. For this and other reasons, z_r = sqrt(1 - 2 G M / (r c^2)) cannot be the exact relation. So now we will examine further what the relation should be.
    The locally measured speed is only for a freefaller starting from rest at infinity, this is not the general speed of any freefaller. In deriving that result it is necessary use the fact that he starts from rest.

    To get the correct result for any initial velocity, notice that is a conserved quantity. Since we have him starting from infinity we take . Now because he starts from rest, at infinity we have , and thus , and plugging that into the schwarzschild metric gives .

    However if we let him start with an initial velocity at infinity, then at infinity , where the gamma is the SR time dilation between the freefaller and the observer at infinity, so to get the result you then need to plug in in the schwarzschild metric and solve for . Where is the time for a stationary observer.

    ETA: what this well get you is where is the initial time dilation due to the initial velocity at infinity. If that velocity is zero then it will cancel with the other 1 and get you your formula for the speed of a freefaller.
    Last edited by caveman1917; 2012-Apr-11 at 02:22 PM.

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    Quote Originally Posted by grav View Post
    Thanks, but I'm not sure what you are saying about rapidity. What I am presenting actually does not refer to rapidity or speeds at all for that matter, but determines gravitational time dilation in respect to stationary point masses at different locations to an observer as z_ total = z_{m_1, r_1} * z_{m_2, r_2} * z_{m_3, r_3} * ... , which as derived from the example I gave and considering the low order approximation, works out to z_{M, r} = 1 / e^(G M / (r c^2)) for each point mass and the same for a spherical mass of uniform density.
    http://en.wikipedia.org/wiki/Rapidity

    Caveman's results can be attained by treating the 2GM/rc^2 as a boost in rapidity from infinity and using the special relativistic rapidity addition formulas.

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    Thanks again, guys. The beginning of the post, however, is not what I am presenting, only giving a quick, perhaps over-generalized retrospect of GR and a couple of concerns I have about it. The last part of the post, not having anything to do with freefallers or speeds or rapidity, only the time dilations of hovering observers near point masses and how they relate to what the distant observer measures, is what this thread is about. I appreciate your posts though. I'm sure they'll come in handy further down the road. As for the variables in the Schwarzschild metric being exact solutions, they may be as derived from the vacuum energy solutions, or EFE, I don't know as I couldn't really follow them, but I thought I had read that it was only the general form of the metric itself that is said to be exact (basically just the Minkowski metric with variables for the time dilation and radial length contraction) but the variables themselves would depend upon the full theory of gravity. Maybe I misread it.

    Anyway, though, the thing is that with the last part of the post I presented, it appears I became too focused upon those time dilations and neglected one very important thing, the radial length contraction. The general relation z_total = z_a * z_b * z_c * z_d might still apply for the time dilation the distant observer measures in respect to the local values, but with radial length contraction, the distances to each of the masses will be different from what the local observers measure, so the distances according to the local observers would have to be changed accordingly for what the distant observer measures to gain the same relation for the time dilations in respect to the distance to each mass. The length contraction would depend upon the contributions of each of the masses and the angles between them and the observer, which complicates things immensely and makes GR non-linear. In order for my conclusion to stand up as it is, we would have to postulate that there is no length contraction within a gravitational field, but I have nothing upon which to base this.

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    It appears that you are not presenting anything against the mainstream, but observing that the Schwarzschild metric is non-linear, and that the length contraction is not isotropic.

    The speed of light is not the same in the radial and transverse directions using the Schwarzschild metric.

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    Quote Originally Posted by utesfan100 View Post
    It appears that you are not presenting anything against the mainstream, but observing that the Schwarzschild metric is non-linear, and that the length contraction is not isotropic.

    The speed of light is not the same in the radial and transverse directions using the Schwarzschild metric.
    Well, it definitely started out ATM. Maybe not so much now unless we postulate no radial length contraction, which would bring us back to the conclusion of the original ATM. I believe your work postulates no radial length contraction, doesn't it? If so, then this is the time dilation in a gravitational field that would directly apply to that.

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    Quote Originally Posted by grav View Post
    Well, it definitely started out ATM. Maybe not so much now unless we postulate no radial length contraction, which would bring us back to the conclusion of the original ATM. I believe your work postulates no radial length contraction, doesn't it? If so, then this is the time dilation in a gravitational field that would directly apply to that.
    Without this radial length contraction, gravity does not match observation as a purely metric theory. My work should stand alone in its own thread, but it also includes a non-metrical mechanism to account for this effect, which has been verified relative to GR.

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