General relativity finds that gravity dilates the wavelength of light and the time period of an atomic clock. What does gravity do to the length of a (material) rod? If you know the answer, please cite me a reference or two.
In GR gravitation potential dilates time; In special relativity velocity dilates length and time. See Einstein online...of wikipedia articlesd on relativity. All because it is the speed of light that is fixed, not time nor space. "What does gravity do to the length of a (material) rod?" Within a black hole, it crushes it to nothingness.
Special relativity is not in discussion here. ln gereral relativity, here is how the frequencies of a vibrating particle are related: (frequency at infinity) = (1 + k.m/r) (frequenct at r), where k is modified gravitational constant. You can deduce dilatation of light wavelength and clock time period from the above. Is that eq. true for a rod's length?
You are no doubt feeling some frustration without any satisfactory response forthcoming. Has been my battle too - but have come to realize that in GR spatial distortion is a bit of a slippery eel thing. 'Go down there' to check and you shrink/bend/whatever same way as the rod, 'stay out here' and a host of issues re measurement and what it really means crop up. There is a short answer that may or may not satisfy you right here in PF: https://www.physicsforums.com/showthread.php?t=145285 A more lengthy and technical discussion, : https://www.physicsforums.com/showthread.php?t=404153 People seem far more comfortable in general with net results - just aply EFE's and out pops orbital dynamics etc, rather than thinking in terms of a 'clean' partitioning into time vs distance effects.
Thank you, Q-reesus for a fitting answer. I will check the sites you suggested. Meanwhile, if you are interested in knowing how I derived that in GRT, a rod is longer closer to mass, please let me know. I can't post it here unless I have the Forum's permission. I have been asking it because in SRT, a moving rod appears contracted in the direction of motion to an observer. Thanks again.
SinghRP, the following thread, where Q-reeus and I have been discussing the effects of spacetime curvature around a gravitating body on radial vs. tangential distance measurements, might also be worth a look: https://www.physicsforums.com/showthread.php?t=541317 It's also worth noting that there are some general issues with the way your question is posed, which may be of interest: (1) What is the rod's state of motion? Usually, unless otherwise specified, it is assumed that the rod is "hovering"--that is, that it remains at a constant radial coordinate r. But this requires the rod to accelerate, in order to remain at a constant height against the pull of gravity. So if you want to see any effects of spacetime curvature on the rod's length, you have to first factor out the effects of the acceleration. (2) Also, the usual argument for "length contraction" of a rod placed radially in a gravitational field is based on the metric coefficient g_rr in Schwarzschild coordinates being greater than 1, and getting larger as the radial coordinate r gets smaller. But this is not true in other coordinates; for example, in Painleve coordinates, g_rr is 1. This means, for example, that a rod freely falling towards the gravitating body, instead of "hovering" at a constant radial coordinate, will *not* see any "length contraction" even if the usual argument is correct. (3) How is the "length" of the rod to be measured? The "obvious" way to do it is to make marks on the rod while the rod is very far away from all gravitating bodies, so spacetime in its vicinity can be assumed to be flat, and then measuring the distance between the marks once the rod is lowered into the gravity well and placed radially. But this begs the question, how do we measure the distance between the marks? We could use another rod, but the same question would arise for it. Or we could use light, say a laser rangefinder at one end of the rod and a mirror at the other, but gravity affects light too. There is simply no "distance measure" we can use that is unaffected by gravity, so whatever effect gravity has on the rod, it will have on the distance measure too. (4) There is another way to judge the rod's length that avoids the above problem, by measuring stresses in the rod (or some other frame-invariant observable). For example, I make marks on the rod in some region of spacetime that is practically flat, after verifying that there are no measurable stresses in the rod. Then I place the rod alongside a second identical rod with identical marks on it, also unstressed. Then I place the first rod under known stresses, and measure its marks against the marks on the second, unstressed rod. This allows me to calibrate the rod's stress-strain ratio--i.e., how the distance between its marks changes with stress. Now, having done all this, I lower the rod into the gravity well and place it radially. But GR predicts that, when I do this, I will measure *no* stress on the rod! (Strictly speaking, I will measure some stress due to the acceleration the rod has to sustain to "hover" at a constant radial coordinate, but we already talked about that above; once we correct for that, there will be no measurable stress in the rod. There may also be effects of tidal gravity, which I don't think we need to go into detail about here; they would need to be factored out the same way acceleration is.) When people in those various threads that were linked to above talk about there being no locally measurable effects of the difference in metric coefficients (radial vs. tangential, in Schwarzschild coordinates), this is the sort of thing they're talking about: whenever you look at a frame-invariant, physical observable, locally, you find that there is *no* difference radial vs. tangential. It's also worth nothing that the case of "time dilation" in a gravity well is somewhat different, because there is a frame-invariant physical observable we can use to measure it--the gravitational redshift/blueshift between observers "hovering" at different heights in the field.
Peter..some interesting comments above.... So can we place identical light sources on the ends of a radially placed rod in the vicinity of a mass and then observe the redshift (say) from a distant observational point...to measue distance?? Singh.... I don't understand what that means....You know that special relativity is a special case of GR I am sure....But questions of distance (length) get less precise in curved spacetime. Other "tricks" to gain some insight would be Einstein's equivalance principle and consideration of uniform gravitational fields (no tidal effects). Are you familiar with " Born Rigidity" ?? Good read....from prior post above: A more lengthy and technical discussion, : https://www.physicsforums.com/showthread.php?t=404153
Basically, the effect of gravity on space-time is that it change the metric which characterize the property of space-time. In Newtonian approximation the time component of metric contribute to potential the most, which is affecting the wavelength. But I think that it is based on approximation of relatively flat space, would there be effects on space component as well?
Neglecting possible higher order derivative effects, what is written above is of course a logical necessity simply because, as I put it in #4 "'Go down there' to check and you shrink/bend/whatever same way as the rod," If I'm not mistaken SinghRP understands that bit, but like myself is trying to get a handle on relating, in some physically meaningful way, effect of gravity on 'length down there' as it relates to us 'out here'. Bottom line - if curvature of spacetime has any real meaning for space as well as time (the uncontroversial redshift part), there must be evident effects observable to us 'out here'. And not just bending of light. For instance, there must be some sense in which say a neutron star will appear, after all mechanical stress/strain, gravitational lensing etc, is factored in, to have a different diameter based on 'warped length scale' - a geometric effect independent of any coordinate system used. So, the statement 'neutron star xyz has a diameter of 20km' refers to what - local measure or our measure? It was argued elsewhere that while something will be there, too much inherent 'freedom of measurement choice etc' for definite, unambiguous predictions. The slippery eel.
Remember that in the passage you quoted I was measuring distance by measuring stress. Stresses are invariant physical observables. So the "shrinking/bending" of the measuring device is irrelevant. I was making an even stronger statement: in terms of stresses, there is *no* "shrinking/bending" of measuring devices "down there" relative to "up here". The only effects of the K factor "down there" are *how many measuring devices* will fit between concentric spheres, relative to the Euclidean prediction. I don't know how I can make it any clearer. It has no effect on *length*, per se. It has an effect on "how much length" is present, radially, in between concentric spheres, relative to the Euclidean prediction. That *is* a physically meaningful way of describing the difference between "down there" and "up here". The K factor, as I defined it, is a physical observable. I told you how to physically measure it. There are; the K factor as I defined it is just as observable "out here" as "down there". The non-Euclideanness of packing of little objects can be observed from anywhere. You would have to quote a specific instance of such a statement to answer this, and the answer would depend on the specific instance. As "diameters" are usually quoted, I suspect they are actually in "radial coordinate units", meaning they are really the square roots of areas divided by 4 pi, or they are circumferences divided by 2 pi (well, for "radius", not "diameter", but it's easy to relate the two). If actual "physical diameter" is meant, then yes, you would have to evaluate the K factor throughout the interior of the star to see how much actual, physical distance there was inside a sphere with the area of the star's surface, given the K factor.
My response to Q-reesus and PpeterDonis at Post 835. Thank you for the dialogue. Let’s keep the issue simple. The issue with length change in gravitational field has been on my mind ever since I read Adler, Bazin, and Schiffer’s “An Introduction to General Relativity” and George Gamow’s “Gravity” – that was a long, long time ago! I searched several books, journals, and websites. I did not find satisfaction. The clearest statement I came to was what Gamow wrote in simple English: “A typist working on the first floor of the Empire State Building will age more slowly than her twin sister working on the top floor. The difference will be very small, however; it can be calculated that in ten years the girl on the first floor will be a few millionth of a second younger that her twin on the top floor. In the difference in gravity between the surface of the Earth and the surface of the Sun, the effect is considerably larger. A clock placed on the surface of the Sun would slow down by one ten-thousandth of a percent in respect to the terrestrial clock. Of course, nobody can pace a clock on the surface of the Sun and watch it go slow, but the expected slowdown was confirmed by observing the frequencies of spectral lines emitted by atoms in the solar atmosphere.” These statements are the clearest even for a physicist like me. In addition, it isolates gravity from the clutter of issues. I knew that light produced on the sun is redshifted compared to the light produced at infinity. [Redshift: the wavelength is elongated. This is not Doppler or Wolf shift.] So, I asked myself: “Does the girl on the first floor get taller or shorter relative to her twin on the top floor?” I will force myself now not to worry about the non-uniformity of gravitational field over the girls nor about whether they are accelerating or in relative motion to each other. So, I stared with very basic physics and mathematics and considered to ignore all perturbations and clutters. I took Einstein’s equation about the frequencies (ν) of a vibrating particle from his book “ The Meaning of Relativity”: ν∞= (1 + κm/r) νr , where m is mass under whose gravity the vibrating particle is, r is the distance between the two, and κ = G/c2, where G is the gravitational constant. Then I imagined a very thin wire, whose “particles” (charge e, mass δm << m) are separated by x. Under the electrostatic forces of its neighboring particles, each particle oscillates with a frequency. (Heisenberg’s uncertainty principle forces the oscillations.) The frequency of this oscillation can be derived elementarily to be: f = z/xq. Here z and q are parameters of the rod. For instance, for a one-dimensional rod dimensional rod, z = (1/π) (Q e2 /δm)1/2 and q = 3/2. Here Q is Coulomb’s constant. Substituting the particle’s frequency into the Einstein’s equation, we have: xr = (1 + κ m/r)1/q x∞. This tells me that: the rod is longer closer to the mass; and the rod is flattened and disintegrated near a black hole (m/r → ∞). If the gravitational field is not uniform over the rod, the rod is deformed. Now I answer my question: “The girl on the first floor gets taller relative to her twin on the top floor.” So this is my “short” contribution to the dialogue. I don’t see any violation of basic physics in above. I hope I found a satisfactory answer to my original question. [By the way, I already moved to the first floor.] What do you think? Please keep the discussions going on this topic.
SinghRP: If you can get a hold of a copy of 'Gravitation and Relativity' - M.G.Bowler. Chapter 6 'The distortion of reference frames' goes into all that in what some might consider an unorthodox way. He seems to implicitly work in isotropic coordinates, and references *all* locally gravitationally effected quantities to their coordinate values. He obtains the opposite finding - shortening. I'm sure someone here will blast the approach taken there, but the results do give a consistent outcome. Typsetting is attrocious btw.
I think this formula, as you give it, is approximate; it assumes that km << r. But that's only a technical point, and doesn't affect the argument; I just mention it in passing. The more important point is that this equation relates *observed* frequencies at one point, to *emitted* frequences at the other. For example, if someone shines a laser emitting light of a known frequency from one point, the light will appear to have the frequency given by the equation at the other point. We say informally that light descending into a gravity well from far away is "blueshifted" (it's observed to have a higher frequency than the known frequency of emission) when it's observed at radius r, and light ascending out of a gravity well from radius r to someplace far away is "redshifted" (it's observed to have a lower frequency than the known frequency of emission). So this equation doesn't say anything about the "real" frequency at the point of emission; it only talks about how the signal carrying information about the frequency is changed as it passes through the intervening space. How is x measured? Remember all the difficulties I raised in my last post about that. How are you deriving this from Heisenberg's uncertainty principle? First of all, I don't see Planck's constant anywhere. Second, oscillations due to the uncertainty principle are not the same as oscillations due to Coulomb force. But even supposing, for the sake of argument, that you had a valid equation relating some observed frequency to the distance x between neighboring particles, just plugging that equation into Einstein's equation above about frequencies doesn't mean what you seem to think it means. See below. No, it's not telling you that. It's only telling you that, if the oscillations in the rod generate some sort of signal (for example, oscillations due to electrical forces inside the rod might generate a radio signal--the rod might be a radio antenna), the frequency of that signal if it is observed far away may be different than the frequency the signal had when it was emitted from the source. In order to derive information from the signal about the inter-particle distances x at the source, you would first have to correct for what happened to the signal during its travel, which means you would have to *undo* the effect predicted by Einstein's equation on the signal. The general principle here is that, if you want to use information about frequencies to determine information about distances, you have to use local frequencies--frequencies measured at the same location as the distances you are interested in. It's also worth nothing that nothing about what you have said above involves any non-uniformity of the field *over the rod*. It only involves non-uniformity of the field between the rod and the point where signals coming from the rod are being observed. Non-uniformity of the field over the rod itself is tidal gravity, and I understood that we were supposed to be leaving that out.
To PeterDonis: You have raised good points. I will answer them later. As an old man I take time. Thanks so much for keeping the dialogue alive.
SinghRP: Just to make sure there was no misunderstanding, when I wrote in #13: "Typsetting is attrocious btw." - that was referring to Bowler's book, not your entry! And when I wrote "...references *all* locally gravitationally effected quantities to their coordinate values.", that meant considering the effect of gravitational potential on things like permeability, permittivity, mass, charge, velocity, speed of light, mommentum, even Planck's constant. Not as straight forward as it first seems, but an exhaustive cross checking procedure used there ties it all down in the end.
To PeterDonis at Post 856. Sorry, I am late! I was busy grading papers. My answers follow after yours. I think … in passing. The complete formula is: vx = ((1+km/r)/(1+km/(r+x))vr. In the limit x → ∞, it’s: v∞ = (1+km/r)vr. The more important … the intervening space. You have a point. I will quote Einstein as he interprets the aforementioned limiting formula: “The rate of a clock is accordingly slower the greater is the mass of the ponderable matter in its neighbourhood. We therefore conclude that spectral lines produced on the sun’s surface will be displaced towards the red, compared to the corresponding lines produced on the earth.” Lindsay and Margenau in Foundations of Physics said the same. Personally I think that: Gravity affects atoms and associated emission and absorption of light (A particle vibrates at lower frequency closer to the mass); Gravity affects the passage of light through its field (Light is redshifted as it is rising and blueshifted as it is falling in the field.) How is x measured? Remember … that. There is no need to measure r at great separations. It is avoided by experiments such as Pound and Rebka’s. Physics is replete with overemphasis on measurements. Moreover, coordinate systems and observers are not a part of the laws of nature – E. A. Milne. How are you deriving … Coulomb force. HUP ensures that a particle shall move (Δx must not be zero.) Then, in a crystal, the repulsive forces of neighboring atoms take over the oscillations of the particular atom. (A finger has to prod the pendulum to move. Not at the macroscopic but at the microscopic level, HUP is a very much more effective finger.) But even supposing, … See below. I believe you and I already addressed it previously. No, it’s not telling … on the signal. The general … interested in. As discussed before, gravity does affect vibrators and what they emit or absorb. I am sure we can agree on those for the time being. All I wanted to have at this point is a relationship between the frequency of vibration of an atom and separation distance between atoms in a crystal. (Separations are infinitesimal and local.) So, that substitution is justified. An experiment similar to the Pound-Rebka’s may be conducted, where changes in Bragg’s reflection/diffraction patterns from a crystal (such as NaCl) may show whether spacings between the atoms are changed by gravity. Thermal and other non-gravitational effects must be sorted out. I am very much confident – just being a human here – that atoms in a crystal have larger separations in a stronger gravitational field or closer to a mass. On the other hand, I could be wrong. It’s also worth … leaving that out. The scenario in my statement is quite simple. The separations between the atoms in a crystal are affected differently in a non-uniform gravitational field, so the crystal will be deformed.
Sorry, I am late! I was busy grading papers. My answers follow after yours. I think … in passing. The complete formula is: vx = ((1+km/r)/(1+km/(r+x))vr. In the limit x → ∞, it’s: v∞ = (1+km/r)vr. The more important … the intervening space. You have a point. I will quote Einstein as he interprets the aforementioned limiting formula: “The rate of a clock is accordingly slower the greater is the mass of the ponderable matter in its neighbourhood. We therefore conclude that spectral lines produced on the sun’s surface will be displaced towards the red, compared to the corresponding lines produced on the earth.” Lindsay and Margenau in Foundations of Physics said the same. Personally I think that: Gravity affects atoms and associated emission and absorption of light (A particle vibrates at lower frequency closer to the mass); Gravity affects the passage of light through its field (Light is redshifted as it is rising and blueshifted as it is falling in the field.) How is x measured? Remember … that. There is no need to measure r at great separations. It is avoided by experiments such as Pound and Rebka’s. Physics is replete with overemphasis on measurements. Moreover, coordinate systems and observers are not a part of the laws of nature – E. A. Milne. How are you deriving … Coulomb force. HUP ensures that a particle shall move (Δx must not be zero.) Then, in a crystal, the repulsive forces of neighboring atoms take over the oscillations of the particular atom. (A finger has to prod the pendulum to move. Not at the macroscopic but at the microscopic level, HUP is a very much more effective finger.) But even supposing, … See below. I believe you and I already addressed it previously. No, it’s not telling … on the signal. The general … interested in. As discussed before, gravity does affect vibrators and what they emit or absorb. I am sure we can agree on those for the time being. All I wanted to have at this point is a relationship between the frequency of vibration of an atom and separation distance between atoms in a crystal. (Separations are infinitesimal and local.) So, that substitution is justified. An experiment similar to the Pound-Rebka’s may be conducted, where changes in Bragg’s reflection/diffraction patterns from a crystal (such as NaCl) may show whether spacings between the atoms are changed by gravity. Thermal and other non-gravitational effects must be sorted out. I am very much confident – just being a human here – that atoms in a crystal have larger separations in a stronger gravitational field or closer to a mass. On the other hand, I could be wrong. It’s also worth … leaving that out. The scenario in my statement is quite simple. The separations between the atoms in a crystal are affected differently in a non-uniform gravitational field, so the crystal will be deformed.
The word "non-uniform" is crucial here; for the spacing between atoms in the crystal to be affected, the field needs to be non-uniform *across the crystal*. This is *not* required for the observed frequencies of light emitted by an object to be seen as redshifted from far away; it's only necessary that the field be non-uniform between the point of emission and the point of reception. The field can be uniform across the object that is emitting the light.
Some statements were made previously regarding special relativity. And I said that SR was not in discussion. As you know, SR and GR are two separate topics. Gravity a real force – the fourth fundamental force, assuming there are no others. In GR, gravitation is due to the curvature which matter creates in the field of space-time geometry. The field of space-time geometry is the gravitational field. At the microscopic level, gravitons would be the quanta of gravitational field. No gravitons and gravitational waves have been detected so far. I never knew how matter (creates and) warps space-time! The strong, the weak, and electromagnetic fundamental interactions are mediated by the color, the weak, and electromagnetic fields associated with the color, the weak, and electrical charges of matter or antimatter. There is no such analogy associated with gravitational interaction. I never knew why! SR is an apparent, not real, effect: 1. If we observe a rod moving past us with a uniform velocity v, it will look contracted in the direction of its motion by a factor (1 – v2/c2)1/2. 2. If we observe a clock moving past us with a uniform velocity v, it will appear to be losing time in the direction of motion, its rate slowed by a factor (1 – v2/c2)-1/2. If observer A is moving uniformly with v relative to observer B, then the latter is moving with –v relative to the former. The twin-paradox is not a paradox at all. If an observer is moving with the mesons, s/he sees no changes in their decay rate.