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Natural Science Forum / Physics / Relativity / September 2006Echo time in Schwarzschild coordinates/metric
A very simple question: Suppose, static at a partcular r in Schwarzschild coordinates (in the Schwarzschild metric, of course) I sent a laser pulse radially inward to a mirror 1 meter away (as measured by my meter stick). How do I calculate the time to return by my local clock? Since I use local clocks and meter sticks, I'm tempted so say "2/c". But are any subtleties introduced by the fact that I am using non-inertial coordinates and measuring in the direction of acceleration? What is the order of the correction, if any? > A very simple question: Very simple indeed, so why are you asking? > Suppose, static at a partcular r in Schwarzschild coordinates (in the > Schwarzschild metric, of course) I sent a laser pulse radially inward [quoted text clipped - 5 lines] > non-inertial coordinates and measuring in the direction of > acceleration? What is the order of the correction, if any? What do you think it may be? Don't you think that the mirror will displace to meet the laser? I'll give you a hint. Create an EQUAtion where the intial distance less the accelerated displacement of the mirror equals ct (displacement of the light pulse). From this you can solve for the time it takes for the pulse to traverse to the mirror. Obviously, it is less than 1/c. If you can figure this out, you can figure out the return trip time too. Phil | > A very simple question: | [quoted text clipped - 19 lines] | | Phil We already have one. c = 2AB/(t'A-tA) http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF Androcles > A very simple question: > [quoted text clipped - 7 lines] > non-inertial coordinates and measuring in the direction of > acceleration? What is the order of the correction, if any? I get 6ns Sue... > A very simple question: > [quoted text clipped - 7 lines] > non-inertial coordinates and measuring in the direction of > acceleration? What is the order of the correction, if any? xxein: Suppose r was 2M! Yes, there are subtleties involved exactly because the light will travel through many different inertial frames that are the local limits only derived from the larger dynamic. Each one is slightly different wrt OWLS --- but taken as inertial, TWLS allows a correspondence such as Lorentz and Einstein speculate to great accuracy. Iow , the frames have to be infinitesimal in size to exactly mimic the coordinate transform. But we allow a sloppiness of our measurements to follow the universe's act. Yet, and still, we use a different brand of thinking to realise a gravity (why it is there and how it affects). Our physics is sad until it finds out what gravity exactly is. Yes, we have a supposed metric that is supposed to describe motion wrt mass, but even that runs across limits where the crossing is undefined. All one has to imagine is the variable value of c across different FORs with a moving gravity (non-linear background). Again - if r was 2M? I don't like to say this but I have to. Even with two fairly simple ingredients (mass and energy), there is no predictable outcome for a final product - ever. Nor is there a beginning set. If you would like to imagine no end to existence, why would there have to be a beginning of one? Our universe runs the scope of infinitesimal to infinity - even from our viewpoint. Yet we self-impose a coarseness to this structure to comply to only what we can and have observed. Why? Because that is all we have to work with. We supply limits where there might not be and disallow extensions where there could be. Why? Because we don't think the way the universe operates. Why does gravity exist in the sense of the universe operation? Until you get a good handle on that, all is just human contrivance. Just the way we make a test-theory of how things work. I don't think this was posted. Please forgive a repeat. Edward Green says... >A very simple question: > [quoted text clipped - 7 lines] >non-inertial coordinates and measuring in the direction of >acceleration? What is the order of the correction, if any? Okay, here are the formulas. A light signal directed radially in the Schwarzschild geometry will have coordinate speed 1. |dr/dt| = 1 - 2m/r where m=the mass of the black hole, and units have been chosen so that G = c = 1. Equation 1 can be integrated to compute the change in Schwarzchild time t for a light signal to travel from r_0 to r_1 (where we assume that r_1 > r_0; the reverse direction takes the same length of time) 2. t_01 = r_1 - r_0 + 2m log((r_1 - 2m)/(r_0 - 2m)) For a round-trip, double t_01. Now, for your problem, this doesn't answer the question, because t_01 is not the elapsed time on a clock "stationary" at r_0, and r_1 - r_0 is also not equal to the measured length of a meter stick stretching from r_0 to r_1. We need a couple of more formulas: 3. tau_01 = t_01 square-root(1-2m/r_0) 4. L_01 = Integral from r_0 to r_1 of square-root(1+2m/(r-2m)) dr where we used square-root(1-2m/r) as |g_tt| and square-root(1 + 2m/(r-2m)) as |g_rr|. So to solve your problem, you set L_01 to 1 meter, to find r_1 in terms of r_0. Then plug r_1 into equation 2 to find t_01, and then use equation 3 to find tau_01. If you just want to know the leading correction, then expand everything under the assumption that r_1 is close to r_0. Let r_1 = r_0 + x. Then expand in powers of x and keep terms of order x^2 (terms of first order in x should all vanish). -- Daryl McCullough Ithaca, NY > Okay, here are the formulas. A light signal directed radially > in the Schwarzschild geometry will have coordinate speed > > 1. |dr/dt| = 1 - 2m/r In general, if you know the direction of a light beam and want to calculate its coordinate speed, just take ds^2 = 0. In the case of the Schwarzschild metric and radial motion, you get 0 = (1 - 2m/r) dt^2 - dr^2 / (1 - 2m/r) - r^2 dOmega^2 and setting dOmega^2 = 0, since the light is moving radially, you end up with Daryl's formula. -- Ben [...] Something that has never quite set right with me is distance in GR. Am I right in saying that g_rr tells us about how the radial coordinate changes, and g_tt tells us about how the time coordinate changes? Using g_tt we can find proper time. Can we use g_rr to find a proper distance that corresponds to the length of something as measured locally? Or is the concept of proper distance too much ot a stretch? Eric Gisse says... >Am I right in saying that g_rr tells us about how the radial coordinate >changes, and g_tt tells us about how the time coordinate changes? Not quite. If a clock is sitting at a constant value for r, then the elapsed time on the clock will be given by: delta-tau = square-root(|g_tt|) delta-t So g_tt tells us something about how proper time tau changes as a function of coordinate time t (when r is kept constant). If a clock is travelling so that r is changing (as well as t), then tau is computed by delta-tau = square-root(g_tt delta-t^2 + g_rr delta-r^2) So both g_tt and g_rr can be involved in computing proper time. On the other hand, if delta-t = 0, then we can compute proper length by delta-L = square-root(|g_rr|) delta-r In General Relativity, the more fundamental quantity is ds^2 which is given by ds^2 = g_uv dx^u dx^v This quantity can be positive, negative, or zero. If ds^2 is negative, then square-root(|s^2|) can be interpreted as a proper time, and if it is positive, it can be interpreted as a proper length. The physical meaning of proper time is pretty straight-forward; it is the elapsed time on a clock that travels a spacetime path from position x^u to position x^u + dx^u. The physical meaning of proper length is a little more obscure, though. For very small objects, it is approximately the length of the object as measured in a local reference frame in which the object is at rest. For an extended object, it's not exactly clear to me what the physical meaning of proper length is. >Using g_tt we can find proper time. Can we use g_rr to find a proper >distance that corresponds to the length of something as measured >locally? Both can come into play when measuring either proper distance or proper time. However, you are right, that for an object that is "at rest" (with r=constant), g_tt can be used to compute proper time, and g_rr can be used to compute proper length. -- Daryl McCullough Ithaca, NY > Eric Gisse says... > [quoted text clipped - 30 lines] > be interpreted as a proper time, and if it is positive, > it can be interpreted as a proper length. So if something travels along a time-like path [negative ds^2] we get proper time, and if it travels along a space-like path we get proper length? Zero ds^2 are photons, and either concept is worthless for them. The proper length thing is good to know. > The physical meaning of proper time is pretty straight-forward; > it is the elapsed time on a clock that travels a spacetime [quoted text clipped - 4 lines] > object is at rest. For an extended object, it's not exactly > clear to me what the physical meaning of proper length is. Well at least someone attempts to make that distinction. Proper length gets all due treatment in the special relativity texts I have seen [length contraction and crew], along with proper time. But from what I have seen, nobody talks about proper length in general relativity. I can't decide if the concept is so obvious as to not being worth mentioning, or if the abcense of there always being a local frame large enough to encompass the object clouds the issue enough for it to be swept under the rug. It does raise a question though. I haven't considered it before until I considered proper length in GR. How big would a neutron star be if relativity didn't muck up distances? I don't know how to tackle the question, but I expect the answer to be 'a little bit bigger' because neutron stars tend to be only one order of magnitude away from their Schwarzschild radius. Could you even use the Schwarzschild metric for such a problem? There would be an integration over the interior of the star, so wouldn't there need to be an interior solution? > >Using g_tt we can find proper time. Can we use g_rr to find a proper > >distance that corresponds to the length of something as measured [quoted text clipped - 4 lines] > "at rest" (with r=constant), g_tt can be used to compute proper > time, and g_rr can be used to compute proper length. Nifty. Just saying though, I am aware that the other metric components come into play. I was just using g_tt and g_rr as a simple example. Say a metric that is only a function of r and t. > -- > Daryl McCullough > Ithaca, NY > Something that has never quite set right with me is distance in GR. > Am I right in saying that g_rr tells us about how the radial coordinate > changes, and g_tt tells us about how the time coordinate changes? Not really. g_rr tells how much distance is represented by a (small) change in the r coordinate value. That is, for an infinitesimal increment in r, call it k, the distance between the points with r coordinates r and r+k is: distance = sqrt(|g_rr*k*k|) for the case in which all other coordinates remain constant for the two points. [feel free to notate my "k" with "dr", as is common.] (I use an absolute value to avoid confusion over metric signature.) Similarly, the proper time between two points with t coordinates t and t+p, with p an infinitesimal increment in the t coordinate, is proper_time = sqrt(|g_tt*p*p|) [feel free to notate my "p" with "dt", as is common.] About the only way I can interpret your "how the radial coordinate changes" is along a path: parameterize the path by proper time tau (i.e. I assume a timelike path), then the path is defined by: r = f1(tau) theta = f2(tau) phi = f3(tau) t = f4(tau) for appropriate functions f1(.)-f4(.); these functions of course determine the path (and it determines them). [feel free to notate my "f1(tau)" as "r(tau)", which is common. Etc.] In this case, f1(.) "determines how the radial coordinate changes ALONG THE PATH". Etc. > Using g_tt we can find proper time. Can we use g_rr to find a proper > distance that corresponds to the length of something as measured > locally? Yes, for the case that all other coordinates remain constant for the path along which the distance is to be measured. That is, the spacelike distance is to be measured along a surface of constant t coordinate. > Or is the concept of proper distance too much ot a stretch? No, it is not. But it is not nearly as useful or common as proper time. Proper time is by far the most useful parameterization of a timelike path; proper distance would be similarly the best parameterization for a spacelike path. But timelike paths are those followed by timelike objects, and most of the interesting paths are timelike. While timelike paths are unique, there can be a great ambiguity in spacelike paths, because a spacelike path must exist along some surface of simultaneity, but there is no definitive meaning of simultaneity.... Tom Roberts > [...] > Similarly, the proper time between two points with t coordinates t and > t+p, with p an infinitesimal increment in the t coordinate, is > proper_time = sqrt(|g_tt*p*p|) You are wrong on this. The proper time is dtau = sqrt(ds^2)| / c Where ** ds^2 = g_ij dq^i dq^j Or in its special case, ** ds^2 = c^2 g_tt dt^2 - g_rr dr^2 - g_?? d?^2... > [...] > [quoted text clipped - 5 lines] > path along which the distance is to be measured. That is, the spacelike > distance is to be measured along a surface of constant t coordinate. Wrong. See above. > > Or is the concept of proper distance too much ot a stretch? > [quoted text clipped - 6 lines] > because a spacelike path must exist along some surface of simultaneity, > but there is no definitive meaning of simultaneity.... There is no concept of proper distance under GR. >> [...] >> Similarly, the proper time between two points with t coordinates t and [quoted text clipped - 5 lines] > Where > ** ds^2 = g_ij dq^i dq^j That is precisely what I wrote, for the situation I am considering, using the notation I defined in that post (with c=1). <shrug> [I'm ignoring the extraneous "|" in your first equation.] Tom Roberts > > You are wrong on this. The proper time is > > dtau = sqrt(ds^2)| / c [quoted text clipped - 5 lines] > > [I'm ignoring the extraneous "|" in your first equation.] This is not what you wrote. You wrote > Similarly, the proper time between two points with t coordinates t and > t+p, with p an infinitesimal increment in the t coordinate, is > proper_time = sqrt(|g_tt*p*p|) Yes, please ignore the extra "|". It is my mechanical mistake in which I had originally written the said parameter as a function of absolute value, but since then I had changed my mind. Instead, I chose to express the absolute value in terms of the square root of the squared of the said quantity. In doing so, somehow due to my stupidity I did not erase everything pertaining to that absolute value thing. However, your mistake is not a typo like mine. It is a reflection to your deteriorating understand of physics. It shows in the past few months ever since you quit your engineering job. A few months ago, I am sure you would not have made this fundamental blunder. Or maybe it is I who as you have accused many times over that my reading comprehensive is very poor. [...] This is a lost cause. Admit you f.cked up and move on. > [...] > > This is a lost cause. Admit you f.cked up and move on. They never admit they f.ck up. It takes guts - which they haven't got. They rather be exposed as dishonest rats than as a person like you and me, capable of making elementary mistakes. That is how they work. That is why they try to hide their identities. You should know that by now ;-) Cheers, Dirk Vdm , > but there is no definitive meaning of simultaneity.... > > Tom Roberts The probability that one does not know ones identity is dern near zero. Contact Dirk Van de Pee Pee for wagers you don't ever belerive your can collect on. $AU1,000 http://www.esa.int/SPECIALS/GSP/SEM0L6OVGJE_0.html http://einstein.stanford.edu/ Signed Not Dennis... ROFL http://www.google.com/search?hl=en&lr=&safe=off&q=%22dennis+mccarthy%22++usno&bt nG=Search http://www.zetatalk.com/theword/tword17u.htm http://www.leapsecond.com/ ... Sue... "Incident Wave Impedance" http://www.conformity.com/0102reflectionsfig3.gif http://www.conformity.com/0102reflections.html "Retarded potential" http://farside.ph.utexas.edu/teaching/em/lectures/node50.html "Visualizations" http://web.mit.edu/8.02t/www/802TEAL3D/teal_tour.htm Fantasy: http://www.cs.cmu.edu/~rgs/alice-VII.html http://casa.colorado.edu/~ajsh/schwp.html > Edward Green says... > > [quoted text clipped - 49 lines] > Daryl McCullough > Ithaca, NY Let's see if I get this right, (I'll clarify Ed's gedanken a bit). Gedanken: Suppose I erect a tower (~1 km tall), with observer K' on top and K at the bottom. Either observer may be equipted with a radar ranger - reflecting from end to end - to produce a *standing wave* with "n" wavelengths. Each observer using interferometry confirms the wave is standing and can confirm the invariant "n" by technical means. K' measures the frequency of the wave to be f' and it's wave- length to be z', likewise K measures f and z. K' computes the height of the tower to be Z' = n*z' and K finds Z = n*z. The "red-shift" increases the wavelength of z' relative to z, hence K' measures the tower to be taller than K does, and thus K measures the tower to be shorter, in proportion to the well known "rate of time" differential caused by the gravitational potential in accord with Quantum Theory's Photon Energy = h*frequency. That's a straightforward gedanken to understand why a radially orientated "measuring rod" is "shortened" by the presence of gravitation, as K finds compared to K'. Regards Ken S. Tucker > > Edward Green says... > > > [quoted text clipped - 77 lines] > Regards > Ken S. Tucker That is close. You need to specify clocks at either end that whose nuclear resonance is affected by gravity. R. V. Pound and J. L. Snider, Effect of Gravity on Nuclear Resonance, Phys. Rev. Lett. 13, 539 (1964). [3] The more accurate measurement with Snider. http://prola.aps.org/abstract/PRL/v13/i18/p539_1 http://en.wikipedia.org/wiki/Pound-Rebka_experiment IOW the free space light doesn't shift in frequency. The photons {Planck's) associated with absorbtion emission or hyperfine structure is what shifts. A dielectric or conductive reflector as mentioned in the OP will interact as a linear optic returning the same frequency emitted. RT time: 6ns Sue... > > > Edward Green says... > > > > [quoted text clipped - 80 lines] > That is close. You need to specify clocks at either end > that whose nuclear resonance is affected by gravity. These type of gedanken's assume the "clocks" used are *physically* identical. > R. V. Pound and J. L. Snider, Effect of Gravity on > Nuclear Resonance, Phys. Rev. Lett. 13, 539 (1964). [quoted text clipped - 5 lines] > The photons {Planck's) associated with absorbtion > emission or hyperfine structure is what shifts. I'd agree with that! > A dielectric or conductive reflector as mentioned in > the OP will interact as a linear optic returning the > same frequency emitted. Right, that's why a pair of observers K and K' can clarify the gedanken, it is after-all relativity ;-). >From a clear gedanken, we can analyse the *nodes* of the standing wave along the tower, and find it's non-linear, probably Daryl's log function, and begin to deduce curvature without to much math baggage, which can be caboosed later. > Sue... Regards Ken |
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