Naked Science Forum
On the Lighter Side => New Theories => Topic started by: Mike Gale on 09/02/2017 23:26:41
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The Schwarzschild scaling distance can be interpreted as the distance at which the Newtonian escape velocity is light speed. However, one would expect it to be the distance at which the relativistic escape velocity is light speed. The relativistic distance is exactly one half the classical value, but in order to maintain compatibility with SR in the free fall case (https://youtu.be/1ogiQ2E6n0U (https://youtu.be/1ogiQ2E6n0U)), it is necessary to revise the scaling factor from (1-rs/r) to (1-rs/2r)^2. This follows by recognizing that rs/r=v^2/c^2 in the radial free fall case and (1-v^2/c^2)=(1-GM/rc^2)^2 if you balance kinetic energy against the Newtonian potential using relativistic mass as described here: http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm (http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm)
The scaling factor is then larger by (rs/2r)^2. The resulting metric is not a vacuum solution, but it is approximately so in the weak field limit. Aside from relocating the event horizon and ensuring that the scaling factor does not change sign (thus preserving causality in strong fields), the main consequence of this change is that proper time is well defined beyond the horizon, provided that you switch to the alternate metric signature when the magnitude of the spatial displacement exceeds that of the temporal displacement. This implies a new object, which is complex spacetime - something Penrose has been advocating in the context of his Twistor theory (https://youtu.be/hAWyex1GKRU (https://youtu.be/hAWyex1GKRU).)
This discussion thread is a follow on from Why can't light penetrate the event horizon? (https://www.thenakedscientists.com/forum/index.php?topic=69595.0)
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To get the discussion started, allow me to relay some responses from a GR expert I consulted on this idea:
1) Aside from relocating the event horizon, the physics of the external domain are essentially unchanged. It is possible that the new metric reduces to the old one in some coordinate transformation.
2) Relativistic mass, which is the basis of relativistic escape velocity, is a somewhat obsolete concept and modern GR theorists prefer to think in terms of 4-momentum and rest mass. I don't know if that changes anything.
3) Many people I've spoken to profess the opinion that objects with mass can penetrate the event horizon. If not in the coordinate reference frame, then at least from the perspective of a local reference frame. I maintain that penetration is impossible without QM, but I am at a loss to find any consensus of the issue.
4) Although the Ricci scalar is identical to zero for the both metrics, the Ricci tensor for the new one only vanishes in the weak field limit. The meaning of the Ricci tensor is hard to pin down, but the YouTube Viascience channel (which I highly recommend if your GR or QM skills are at all rusty) describes it in terms of conservation of volume. In that context, I would contest the idea that volume must be conserved in strong fields.
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A related link with a reply from a respected expert: http://physics.stackexchange.com/questions/246133/deriving-a-schwarzschild-radius-using-relativistic-mass (http://physics.stackexchange.com/questions/246133/deriving-a-schwarzschild-radius-using-relativistic-mass).
(No comment on the new metric yet, but he argues that the relationship between the SC scaling distance and Newtonian escape velocity is purely coincidental. He bases that judgement on the premise that the Newtonian radial coordinate is different from the one in the SC metric. I'm not so sure about that because GR would be pointless if it was formulated in anything other than Newtonian coordinates. It's true that the local reference frame is foreign to Newton, but the distant one is most certainly the one on which his equations are based.)
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Another controversial subject related to this matter is the variability of the speed of light in a gravitational field from the perspective of a distant observer. Many people seem to balk at this idea because the mantra from SR is that light speed is invariant. Although GR maintains that constraint in local reference frames, it allows the speed of light between distant points to be variable. Einstein's own treatment of this subject can be found here: http://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22 (http://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22)
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On the subject of SR compatibility, the SC metric (old or new) for free fall (radial or orbital) can be expressed as:
ds^2 = ( c dt / gamma )^2 - ( dr gamma )^2 - ( r d(angle) )^2
Where (Lorentz's) gamma is defined in terms of escape velocity (classical or relativistic, depending on which metric you subscribe to) and proper light speed (c.) Note that escape velocity is equal to coordinate velocity (dr/dt) in the radial free fall case, but not the orbital one. Set ds=0 to find the coordinate velocity of light.
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Another argument I've fielded is the idea that the SC solution is sacrosanct because the scaling distance emerges from the bowels of GR when you constrain it to radially symmetric vacuum solutions. From what I can tell, the form of the SC metric certainly emerges from that constraint, but the magnitude of the scaling distance comes from classical theory (i.e. the weak field limit) and the metrics are equivalent in that domain.
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On the subject of SR compatibility, the SC metric (old or new) for free fall (radial or orbital) can be expressed as:
ds^2 = ( c dt / gamma )^2 - ( dr gamma )^2 - ( r d(angle) )^2
Where (Lorentz's) gamma is defined in terms of escape velocity (classical or relativistic, depending on which metric you subscribe to) and proper light speed (c.) Note that escape velocity is equal to coordinate velocity (dr/dt) in the radial free fall case, but not the orbital one. Set ds=0 to find the coordinate velocity of light.
So with ds set to zero we end up with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2. We are back to Pythagoras.
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I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.
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I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.
Well with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2 we end up with a distance divided by gamma on the left side. On the right is a distance multiplied by gamma. As one expands the other contracts. It's bigger on the inside.
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What do you think that means for the holographic principle and Beckenstein's entropy?
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On the subject of SR compatibility, the SC metric (old or new) for free fall (radial or orbital) can be expressed as:
ds^2 = ( c dt / gamma )^2 - ( dr gamma )^2 - ( r d(angle) )^2
Where (Lorentz's) gamma is defined in terms of escape velocity (classical or relativistic, depending on which metric you subscribe to) and proper light speed (c.) Note that escape velocity is equal to coordinate velocity (dr/dt) in the radial free fall case, but not the orbital one. Set ds=0 to find the coordinate velocity of light.
So with ds set to zero we end up with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2. We are back to Pythagoras.
That's right. Just as in SR, you can recover flat space with a coordinate transform. But it's different from a Lorentz transform because the GR coordinate systems are not in motion with respect to one another. Spacetime dilation in GR is not the same as spacetime dilation in SR. One holds c constant and attributes dilation to relative motion. The other has no relative motion so either dilation is due to variable light speed or variable light speed is due to dilation. Both views are valid. The observables are the same in either case. Whichever view gets you through the night, causality must be preserved if we are going to make any sense of it.
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I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.
I'm still digesting your model, but now that you mention it, a cross reference is in order:
Beginning:
https://www.thenakedscientists.com/forum/index.php?topic=69032.0
Middle:
https://www.thenakedscientists.com/forum/index.php?topic=69585.0
Latest:
https://www.thenakedscientists.com/forum/index.php?topic=69592.0
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I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.
Well with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2 we end up with a distance divided by gamma on the left side. On the right is a distance multiplied by gamma. As one expands the other contracts. It's bigger on the inside.
The same is true of the Minkowski metric if you set x'=vt.
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What do you think that means for the holographic principle and Beckenstein's entropy?
Both should hold in either case because all things (including light) stall at the event horizon.
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If we look at kinetic energy we can say that
1/2 gamma m dv^2 - (GM/dr) (gamma m) = 0
is of interest. If we set a value for the right hand side in a strong field that would be interesting.
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If we look at kinetic energy we can say that
1/2 gamma m dv^2 - (GM/dr) (gamma m) = 0
is of interest. If we set a value for the right hand side in a strong field that would be interesting.
I think you meant: 1/2 gamma m v^2 - (GM/r) (gamma m) = 0
That's not quite right. The correct form is:
gamma * mc^2 - mc^2 = (GM/r) * gamma * m
This reduces to your equation in the classical limit (where v<<c.)
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No that is not what I meant. If you progress through the equation you end up with 1/2 gamma m a^2 - v^2/2 (gamma m). So that the particle has a force moving it but the gravitational field appears to have no such force since it reduces to a kinetic energy term. Which is why people convince themselves that the gravitational force is fictitious.
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No that is not what I meant. If you progress through the equation you end up with 1/2 gamma m a^2 - v^2/2 (gamma m). So that the particle has a force moving it but the gravitational field appears to have no such force since it reduces to a kinetic energy term. Which is why people convince themselves that the gravitational force is fictitious.
Ficticious in the sense of the equivalence principle, but point taken. I still think your equation is flawed because gamma is a function of velocity. You're missing the d(gamma) term.
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Yes in fact it is completely wrong. At the moment I am dealing with lots of stuff so I am not entirely concentrating properly. We currently have no kitchen and chaos reigns. Builders don't appreciate the disruption they cause. My apologies.
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I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.
Well with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2 we end up with a distance divided by gamma on the left side. On the right is a distance multiplied by gamma. As one expands the other contracts. It's bigger on the inside.
Yes - understood.
...and my model's introduction of an additional gravitational time dilation, in relation to GR gravitational time dilation, could, (I think...scratches head), be described with such maths viewed from an alternate perspective that ensures that the distance of a metre remains constant.
Thanks Mike: I haven't published my diagram online, but if it became of interest for you to see it as you further digest, I can pm it to you.
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Yes in fact it is completely wrong. At the moment I am dealing with lots of stuff so I am not entirely concentrating properly. We currently have no kitchen and chaos reigns. Builders don't appreciate the disruption they cause. My apologies.
No worries. I've been there and done that. My advice is lowered expectations.
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Thanks Mike: I haven't published my diagram online, but if it became of interest for you to see it as you further digest, I can pm it to you.
I would appreciate that, thanks.
I'm half way through thread #1. I didn't realize that you have no formal training in physics or mathematics. That explains why some of my arguments seem to fall flat on your ears. I'll keep that in mind when I'm trying to make my point.
If I understand your proposal, you are suggesting that we can eliminate spatial dilation by interpreting the physics of GR and SR in terms of time dilation alone. I think you'll run into trouble with that approach because you'll need the passage of time to be directional in an unaccelerated reference frame. It's essentially the same as the variable light speed approach to SR. You can make the math work with that premise, but you have to inject new laws of physics to explain why time passes more slowly in one direction of space relative to another. Einstein's approach is harder to visualize, but it is actually the simplest option.
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I'm in process of going through my notes to send you a comprehensive description of my representation...
I'm actually having one of those having put something down for a long period of time and finding on picking it up again my education has evolved to understanding what my own self was doing back then much better now moments, which to say so is rather pleasant!
I'll be finished and pm you tomorrow with it.
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Still waiting for the official release of timey's new theory, but I finished digesting the threads (referenced above), I provide my diagnosis here for completeness, even though it is not directly related to the topic at hand:
I think people get confused about the Lorentz transform because each of the various parts of a moving object translates to different places and different times (relative to one another.) Moving observers do not perceive space any differently than the rest of us. They just perceive events to occur at different times and it is the perceived simultaneity of events that defines distances in space. In that respect, you (timey) are not far off base trying to interpret SR and GR in terms of time dilation alone. If you take simultaneity into account, you will find that you are arguing for Einstein's interpretation. No shame there, by the way. The giants of physics made all of these mistakes, too. Your Maxwell quotation is a prime example.
If you want to expand your knowledge of SR and GR, I recommend the YouTube Viascience channel. He tells it like it is in terms we can all understand. Dirty laundry included.
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So in a model of variable speeds for light - which presumably you are placing within a model that is expanding under Hubble's velocity related interpretation of the red shift distance correlation - are you still rendering the measure of a non local metre as variable, as well as non local time being variable?
I'll give the YouTube a go, but find that Susskind's 'theoretical minimum' GR lectures are pretty accessible, and Einstein himself isn't exactly incomprehensible, nor any of the prominent physicists, (who's many books I've read), giving too shabby a description either...
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So in a model of variable speeds for light - which presumably you are placing within a model that is expanding under Hubble's velocity related interpretation of the red shift distance correlation - are you still rendering the measure of a non local metre as variable, as well as non local time being variable?
I'll give the YouTube a go, but find that Susskind's 'theoretical minimum' GR lectures are pretty accessible, and Einstein himself isn't exactly incomprehensible, nor any of the prominent physicists, (who's many books I've read), giving too shabby a description either...
I haven't addressed Hubble expansion in the context of the new metric, but I don't expect it to predict anything new in that regard because the metrics are equivalent in the weak field limit. Furthermore, the equations for Hubble expansion involve some dubious concepts. You have to imagine the Big Bang as repulsive gravity (i.e. a white hole as opposed to a black hole) and then accept the fact that it was a one time deal (a quantum fluke if you will) because we don't see any evidence of it elsewhere. The justification seems to be that the old metric falls silent beyond the horizon so you can imagine almost anything you like in that domain (e.g. reverse causality.) The new metric clamps down on such shenanigans, but it does not preclude the possibility of repulsive gravity.
Glad to hear you're giving Viascience a chance. You won't be disappointed.
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The old metric switches the time dimension for a space dimension in its approach to a black hole.
You, (I think I have interpreted correctly), are saying that the new metric doesn't need to do this, and you suggest changing from the old to the new when moving out of the weak approximation field, of which I can see the advantages, but still remain concerned regarding the fact that if a second and a metre are variables, if one is considering variable speeds for light, what exactly does one hold 'anything' relative to?
But to further analyse what is actually physically occurring for black holes, does it not bother you at-all concerning the second law of thermodynamics, and the conservation of energy law reversal of 'usual' physics with regards to black holes?
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The old metric switches the time dimension for a space dimension in its approach to a black hole.
You, (I think I have interpreted correctly), are saying that the new metric doesn't need to do this, and you suggest changing from the old to the new when moving out of the weak approximation field, of which I can see the advantages, but still remain concerned regarding the fact that if a second and a metre are variables, if one is considering variable speeds for light, what exactly does one hold 'anything' relative to?
But to further analyse what is actually physically occurring for black holes, does it not bother you at-all concerning the second law of thermodynamics, and the conservation of energy law reversal of 'usual' physics with regards to black holes?
Swapping time and space is not so easy because it's not clear how time would get divvied up amongst the 3 dimensions of space. It could be an artifact of the math, just as the Pythagorus solution has a sensible solution and a non-sensical one. Another possibility is the phase of the matter wave overtakes that of the light wave so the object appears to slow down if you continue to apply force after it achieves light speed.
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On 2nd thought, I take that back. Swapping time and space is exactly what's going on. When Richard Feynman was asked how he would describe SR to the lay person, he said "One man's time is another man's space." You can see that for yourself in the Lorentz transform because x' is a function of x and t for example. The Lorentz transform is really a pinch as opposed to a rotation, but Feynman's quip is quite accurate despite his gender bias.
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I truly love Feynman and his intellectual acrobatics coupled with his top sense of humour!
The Feynman Lectures are a blast!
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But you are aware that SR uses absolute time, and relates motion directly back to the standard second via the speed of light?
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I truly love Feynman and his intellectual acrobatics coupled with his top sense of humour!
The Feynman Lectures are a blast!
Agreed. I never met him, but I do enjoy his books. His wit even shines through in his textbooks. (I think that's where I read that quote.) Back to business though. Does it strike anyone as odd that there is no equivalent to the Lorentz transform for the SC metric?
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But you are aware that SR uses absolute time, and relates motion directly back to the standard second via the speed of light?
I think you're misusing the term "absolute time". Physicists take that to mean invariant time, which is to say that it does not depend on your state of motion. The only invariant in SR is the change in distance over time for the crest of a light wave (i.e. light speed.) The variable in SR is the relative velocity of reference frames. GR turns that around by making light speed the variable and reference frame velocity the invariant. (The free fall case is really a sequence of stationary reference frames.)
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Yes - but what SR starts out with as a baseline is invariant time which all SR variable time is held relative to, and SR's use of the speed of light is held relative to a standard second.
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Yes - but what SR starts out with as a baseline is invariant time which all SR variable time is held relative to, and SR's use of the speed of light is held relative to a standard second.
Nope. SR does not define an absolute time. All it can tell you is how my time passes relative to yours. It makes no judgement as to who's right and who's wrong. GR does though. The reference frame of the distant observer (i.e. the free fall reference frame) is absolute in that case.
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Hmmm... yes that is correct, SR is not defining who is right or who is wrong...
But from which rate of time does SR start out from when it calculates the difference between your rate of time and mine?
Surely the difference between your rate of time and mine is arbitrary information if we only recognise the difference and haven't started out from the position of 'knowing' what rate of time either of us are operating at?
...and in the case of moving light under the remit of SR, the speed of light 'is' held relative to the standard second.
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That's the "relative" part of special relativity. An observer who is stationary in a moving reference frame perceives the rest of us to be in motion. Aside from the relative velocity of our reference frames, the only thing we can agree on (in the case of uniform motion, which is the "special" part) is the speed of light. You can nominate one reference frame to define a preferred time scale, but the choice is entirely arbitrary unless you invoke GR.
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Clearly - but just saying that's what makes SR special is in as much as saying there is magic afoot.
If people in uniform motion can agree on the speed of light, then they are each holding their own time relative to a standard second.
Placing people in uniform motion in locations of vastly differing gravity potential, what then?
Do all the people in uniform motion in these differing gravity potentials still agree on the speed of light?
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Ah yes - just read your edit.
Invoking GR!!!
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That's the ticket. All observers in free fall measure the same speed of light. It is only when you hold your position (e.g. with a rocket engine or a planetary crust) that you measure a different value. In other words, your perception of time and space depends on your velocity. Your perception of the speed of light depends on how fast you are accelerating.
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But when not in free fall?
Do people in uniform motion in vastly differing gravity potentials agree on the speed of light, where clearly they will be unable to hold the speed of light relative to a standard second, but will have to hold the speed of light relative to the length of second of their gravity potential location?
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They do not agree. The speed of light on the surface of the moon is different than the speed of light on the surface of the Earth for example, even if you run along the surface of the Earth to keep pace with the moon's orbit. GPS is a prime example of that, although there is a lot of confusion about the magnitude of the effect in the literature.
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Good good.
So - how is it possible to move light across space with SR, when SR is holding the speed of light relative to a standard second?
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Your "standard second" is presumably defined in a reference frame that is stationary with respect to the gravitating mass when the observer is infinitely far away. That's a fair description of the space between galaxies if you ignore the Hubble effect. In that case, the speed of light is c. Light is going as fast as it possibly can. Barring the exterior of white holes and the interior of black holes, there is no reference frame in which it can go any faster.
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I don't like 'presumably' or 'infinity far away' much... but oh well for the mo.
Going back to moving mass across differing gravity potentials with SR.
When SR makes its length contraction calculations we are talking about 0.806c (might have got that figure wrong but can't be arsed to look it up) causing a 50% length contraction.
But if the length of second in a reference frame is differing from a standard second, then the percentage of the speed of light will not be 0.806 and the length contraction will not be 50%!
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If you're talking about the NIST second, it is defined in terms of the frequency of oscillation of a cesium 133 atom on Earth (at sea level or in free fall. I can't remember which.) The vacuum speed of light, which is that measured by an infinitely removed observer or any other observer in free fall, then defines the standard meter. The cesium frequency is an arbitrary choice, but it's a good one because it's very stable in any given reference frame.
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According to SI units the metre is defined as per the speed of light relative to the SI unit of the standard second, as per the oscillation of a caesium atom.
The distance of a metre 'is' now actually defined by the speed of light.
It is true that a caesium atomic clock can be calibrated to oscillate to the tune of a standard second at any gravity potential, but once calibrated, a cession atomic clock will oscillate at a higher frequency placed in a higher gravity potential, and at a lower frequency when placed in a lower gravity potential.
So - when moving mass across differing gravity potentials, the speed of light will, if the equivalence principle is upheld be differing because of the differing length of second, and the percentage of the speed of light that the mass is moving at in reference frames of differing lengths of second will be changing, and the length contraction will be altered, won't it?
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It is of interest that the caesium atomic clock has only been tested under lab conditions*, and that those lab conditions have been restricted as to the gravity potential of the co-ordinates of the lab itself...
(*Edit: Although caesium atomic clocks are (I think) used on satellites with respect to GPS, the actual rigorous testing of gravity potential considerations have been conducted under lab conditions.)
Quantum clocks are far more sensitive than the caesium atomic clock, but are so far unreliable and have to be used in conjunction with the caesium atomic clock in order to be of any use.
The caesium atomic clock itself 'can' be more sensitive - but the physical dimensions of the clock, in that the atoms are being processed in a 3 foot high chamber, negate any further sensitivity because the atoms will change frequency within their movement in the dimensions of that chamber. Therefore the caesium atomic clock is 'dumbed down' by the choice of microwave frequency used to excite the caesium atoms by passing them through the microwave beam on their upward trajectory.
NIST do have plans for a portable caesium atomic clock, which I am waiting for with bated breath. I am almost sure that it will be possible to standardise a second at sea level earth and achieve an absolute that can be used to relate the difference any different gravity potential may have against.
I also am very interested in whether or not a geological difference in mass density will affect the clock if 2 clocks were to be placed in the same gravity potential, but at locations of vastly differing geological density...
It is my thoughts that gravity potential may affect a clock in one way, but that the difference in mass density will affect it oppositely... where the clock in the denser location will run faster.
Going back to SR considerations being altered by an invocation of GR in relation to the equivalence principle... Do you have any thoughts?
Where I have run through the scenario myself, (not mathematically, but geometrically), I encounter dimensional problems...
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Wait a tick. The frequency at which a cesium atom oscillates depends only on the local field strength (electromagnetic and gravitational) and temperature. It's not like a quartz clock, where the frequency of oscillation is controlled by applying pressure to the crystal. It is an oscillation between two ground states, which have equal probability in a benign environment. Each transition releases a photon, which can be detected. The elapsed time between detection events defines the second, not the other way around. From the perspective of a distant observer, frequency decreases with increasing field strength so a local second as perceived from afar is longer if the clock is experiencing gravity. This is evident from the metric because dT=dt/sqrt(1-rs/r) if dr=r*d(angle)=0. GPS compares clocks in free fall with clocks on the ground so there can be no doubt about the validity of this model. There have been several experiments with clocks at elevation, too. In fact, standard time is routinely corrected for variations in elevation due to seismic activity.
I don't know what you mean by "quantum clock", but that would be a fair assessment of a cesium clock. That is, they are one and the same. Again though, I don't understand your obsession with the "standard" second. In the absence of gravity, there is no preferred reference frame. The choice is entirely arbitrary and has no impact on the physics.
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And again, I must insist that we focus this discussion on the topic at hand. If you want to debate concepts like the standard second, you (timey) should open another thread. Send me a PM if you want my input. I'm happy to oblige. I find these kinds of discussions instructive (to myself) because it forces me to formulate my thoughts in non-technical terms. I think it was Einstein who said something like, "if you can't explain it in everyday terms, you don't understand it yourself."
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To that end, there are (I think) 3 topics on the table:
1) Implications of the new metric (e.g. for Big Bang and Hubble.)
2) Implications of complex spacetime (e.g. for QM.)
3) The form of the coordinate transform for a GR metric.
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I think Flamm's paraboloid is the right way to think about the coordinate transform:
t'=t*sqrt(1-rs/r)
r'=r/sqrt(1-rs/r)
angle'=angle
Does that make sense? I keep getting confused between the local free fall frame and the local stationary frame. I think these transforms apply to the latter. Unlike the Lorentz transform, there is no mixing of time and space (because the reference frames are stationary with respect to one another.) The equivalent for the new metric is:
t'=+/-t*(1-rs/2r)
r'=+/-r/(1-rs/2r)
angle'=angle
Where the sign is positive for r>rs and negative otherwise. I guess that would preclude complex spacetime. Bad news for Penrose.
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I think the white hole scenario corresponds to the negative of the scaling distance. Does that make sense? Any Big Bang aficionados out there?
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Wait a tick!
Very slick!
And again, I must insist that we focus this discussion on the topic at hand. If you want to debate concepts like the standard second, you (timey) should open another thread. Send me a PM if you want my input. I'm happy to oblige. I find these kinds of discussions instructive (to myself) because it forces me to formulate my thoughts in non-technical terms. I think it was Einstein who said something like, "if you can't explain it in everyday terms, you don't understand it yourself."
The choice of which question you choose to answer might be having a bearing on the 'focus' me thinks.
For instance in relation to this:
Back to business though. Does it strike anyone as odd that there is no equivalent to the Lorentz transform for the SC metric?
You might have chosen to answer this:
Going back to moving mass across differing gravity potentials with SR.
When SR makes its length contraction calculations we are talking about 0.806c (might have got that figure wrong but can't be arsed to look it up) causing a 50% length contraction.
But if the length of second in a reference frame is differing from a standard second, then the percentage of the speed of light (edit: that the mass is travelling at) will not be 0.806c in that reference frame, and the length contraction will not be 50%!
And with respect to this:
I don't understand your obsession with the "standard" second. In the absence of gravity, there is no preferred reference frame. The choice is entirely arbitrary and has no impact on the physics.
You might consider this:
I am certain that our level of understanding has a lot of room for growth. There are a lot of known unknowns, but almost surely more unknown unknowns.
My enjoyment of physics is in exploring the known unknowns, and the unknown unknowns.
My fascination with the standard second:
https://www.thenakedscientists.com/forum/index.php?topic=69800.0 (https://www.thenakedscientists.com/forum/index.php?topic=69800.0)
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I think my coordinate transform is wrong. It has to satisfy this line element:
(cdt')^2 - (dr')^2 = (c'dt)^2 - (dr)^2
The primed coordinates are those in the Lorentz transform and c' is the coordinate speed of light in the SC solution:
cdt'=(c'dt-vdr/c')*gamma
dr'=(dr-vdt)*gamma
c'=c*(1-rs/r) for old metric or c'=c*(1-rs/2r)^2 for the new one.
Note that gamma is the same in both reference frames because v/c'=v'/c.
The equation reduces to the Minkowski metric when rs=0 but in order for it to reduce to the SC metric when v=0, gamma has to morph into 1/sqrt(1-rs/r) so that:
cdt'=c'dt/sqrt(1-rs/r)=cdt*sqrt(1-rs/r)
dr'=dr/sqrt(1-rs/r)
That seems to do the trick, but plugging these values into the line element gives the coordinate speed of light. I did not expect that to happen, but I guess that makes sense. What it's saying is that there is no mixing of time and space in the suspended reference frame. Spacetime is simply squashed in the radial direction. The mixing (i.e. rotation or pinch) only happens when you get moving.
In practical terms, it means that gamma must be interpreted in terms of velocity for the free fall case and scaling distance for the suspended case. Anything in between is a mixture of the two and that's when the local observer feels the force of acceleration. In other words, force is due to the difference between (1-rs/r) and (1-v^2/c^2). The next step then is to express that relationship in mathematical terms.
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Well, that wasn't as hard as I imagined now that I think about. These scaling factors are, after all, just expressions of energy: potential energy in the case of GR and kinetic energy in the case of SR. The old metric ambiguates its roots because it is formulated in a classical context. Multiplying the old scaling factor by mc^2 produces a unrecognizable result:
mc^2 - 2GMm/r
Multiplying the new scaling factor by m^2c^4 gives something more familiar:
mc^2 - GMm/r = mo*c^2
This is the expression for relativistic mass in the free fall case, which is where this discussion thread started. The force one feels when accelerating is due to changes in relativistic mass, which itself is due to the spatial gradient of the potential.
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So we have come full circle. The old metric leads to nonsense when exposed to SR. The new one is perfectly consistent with SR. Can there be any doubt?
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What we have now is yet another new metric:
(cdt')^2 - (dr')^2 = (c'dt)^2 - (dr)^2
where:
cdt'=(c'dt-vdr/c')*gamma
dr'=(dr-vdt)*gamma
gamma=1/sqrt(1-v^2/c'^2)=1/sqrt(1-v'^2/c^2)
c'=c*(1-rs/2r)^2~c*(1-rs/r)
It reduces to SR (i.e. uniform motion) when rs=0 and GR (viz. free fall) when rs/r=v^2/c'^2. The equality does not hold in any other case because, if you're fighting the force of gravity with a rocket engine for example, you are adding energy to the equation. Nobel prizes all around if anyone can come up with a testable consequence of this. Or they might bend that rule if this metric can be related to QM.
See also Re: Relativistic Correction to the Schwarzschild Metric (https://www.thenakedscientists.com/forum/index.php?topic=69764.msg508226#msg508226)
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With regards to QM, the line element must be related to the argument of the exponent in the De Broglie wave equation:
Psi=exp(i*mv/hbar*(x-vt))
The idea would be that the line element represents the magnitude of a complex vector:
dz=dr+ic'dt
dz'=dr'+icdt'=(dr-vdt)*gamma+i(c'dt-vdr/c')*gamma
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Crawling on all fours round your maths as a toddler who has not yet learned to walk, (or talk), could I please just ask you this Mike:
You see where you say about this hbar being related to vacuum permittivity - is vacuum permittivity relating to 'spatially' variable spaces in space?
Edit: Where Mike has now edited his post, it will be extremely difficult to see how my question relates!
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Vacuum permittivity (eo) and permeability (uo) arise in the context of the Coulomb force. They represent the elasticity of spacetime and are completely defined in terms of light speed and pi:
uo=4*pi/10^6
eo=1/c^2/uo
It's analogous to waves on a guitar string, for which the speed of propagation depends on string tension. The seemingly arbitrary definition of uo is an artifact of the units of measure.
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I removed the angular coordinates from the metric in my last edit to Reply#57 because I realized it's not that simple for orbital free fall. The coordinate speed of light is more complicated.in that case because the velocity vector for the SR correction changes direction over time.
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Yes - that is so! ... it will change over time and distance.
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I think we can recover simplicity by switching back to Cartesian coordinates:
(cdt')2 - dx'2 - dy'2 -dz'2 = (c'dt)2 - dx2 - dy2 -dz2
The complexity is then in the coordinate transforms because the speed of light depends on the direction of motion.
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So's not to clog up Jeff's thread as we are getting off topic there...
Well since relativistic effects have been experimentally confirmed at speeds under 30 miles an hour, then I think we can ditch the classical approach
And if it is momentum that changes with v = a, then in the case of free fall surely the energy change in momentum is due to that which is the cause of the acceleration, and not that which is being accelerated.
It's a bit of both. Mass increases with velocity and velocity increases with acceleration. That's why GR theorists prefer 4-momentum. It reminds them that relativistic mass is a consequence of spacetime dilation. It's typical insider terminology though. They want to make you feel stupid if you prefer to think in terms of relativistic mass. It's so passe.
Space time dilation of space, or space time dilation of time?
Both. That's what SR teaches us.
So when you employ coordinate speeds for light that's GR time dilation related?
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So when you employ coordinate speeds for light is this GR time dilation related?
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Yes. SR dilation is due to velocity. GR dilation is traditionally thought of as a mixture of velocity and acceleration, but I contend that it should be exclusively acceleration.
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So - if your coordinate speed of light is due to GR time dilation, then the percentage of the speed of light that m's velocity is, as per held relative to a standard second, will be decreasing.
Therefore the SR time dilation that is caused by your velocity will also be decreased, as per that reference frame.
But where the fu*k are you?
SR length contraction and the reciprocal dilation of space will render your coordinates, and therefore the relevant coordinate speed of light difficult to define!
Correct?
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I'm going to reformulate what I've said last post a little more coherently:
If your coordinate speed of light is due to GR time dilation - then in the reference frames where the speed of light is held relative to the shorter seconds at h from M = Earth (longer seconds for inbound to BH), m's constant velocity as per held relative to a standard second* will be a decreasing (increasing for BH) percentage of the reference frame that m is passing through's speed of light, and this will affect the SR measurements.
*Unless you state m's speed relative to the shorter second (longer second for BH) of the reference frame, in which case m's speed as per a standard second will be increasing (decreasing for BH), also affecting the SR measurements.
SR holds all of its time and length measurements relative to the speed of light per standard second - but by the remit of a speed of light held relative to the shorter seconds (longer seconds for BH) of the reference frames m is passing through, the SR time dilation that m will be experiencing, (or appears to be experiencing, dependent on your school of thought*) is decreasing (increasing for BH) as the speed of light of each reference frame is held relative to shorter seconds (longer seconds for BH)
(This being true unless you have decided that m's speed is increasing (decreasing for BH) as per the standard second when passing through reference frames with shorter seconds (longer seconds for BH))
*In which case, as per the notion that the shorter seconds (longer seconds for BH) of GR time dilation are affecting m's speed, is the speed m is travelling at also affected by SR time dilation?
But whichever you decide, where the fu*k are you?
SR length contraction and the reciprocal dilation of space will render your coordinates, and therefore the relevant coordinate speed of light difficult to define!
Correct?
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Not being able to 'know' exactly where one is in space is what led Katherine G Johnson to utilising Euler's theorem (?) in order to land space shuttle within 20 mile radius, and subsequently launch John Glenn into orbit?
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Locating oneself in a relativistic context is admittedly a tricky business. Bell's spaceship paradox is a prime example. However, the paradox arises from philosophy, not physics. The equations are perfectly self consistent. It is only the interpretation of the result that is contentious. To be more specific, questions of where turn into questions of where and when. Simultaneity is in the eye of the beholder.
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Not being able to 'know' exactly where one is in space is what led Katherine G Johnson to utilising Euler's theorem (?) in order to land space shuttle within 20 mile radius, and subsequently launch John Glenn into orbit?
Probably Kepler. Euler was into fluid dynamics.
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Nope - 'twas Euler's method for definite. I don't make a statement without researching it. Describes curves I believe...
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https://en.m.wikipedia.org/wiki/Euler_method (https://en.m.wikipedia.org/wiki/Euler_method)
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Ah yes. Numerical integration technique. Math, not physics. That came up in the NASA movie "Hidden Figures." It's considered "old school" because computers do that for us nowadays and they can often find exact solutions using symbolic integration techniques.
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Yes that's right. When I first made the post I pulled the term Euler's theorem out of my memory of watching the movie 'Hidden Figures', which is why the word theorem had (?) after it.
Sorry, I realise that saying theorem was a bit misleading. After getting interested by the memory and further investigating, I corrected myself on my own 'my model of a cyclic universe continued again' thread, where I also made mention, but forgot to do so on your thread.
So yes, Katherine Johnson had to resort to using Euler's method when calculating the trajectory of the space mission in order to understand the exact position in space that the shuttle would be located - that it would be then possible to calculate a re-entry and specific landing zone.
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The point being that Johnson's calculation had nothing to do with SR or GR. It was all Newton and the fact that she had to resort to numerical integration techniques is telling both of the power of Newtonian dynamics and of the complexity of the problem. It is a common misconception that classical dynamics is easy compared to relativity. It's not. Relativity may be harder to visualize, but the driving factor for complexity is the number of objects involved in the calculation.
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An example of this is the three body or N body problems.
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That's right. N-body problems are notoriously difficult to solve (for N>2) even before you invoke relativity. I should correct myself on one point though. It is the power of Newtonian dynamics (as opposed to its complexity) that is exemplified by the fact that Johnson didn't need relativity to get John Glenn home.
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Relativity was already a theory when Johnson made her calculation, and time shift was already a phenomenon before Einstein formulated Relativity.
There is no way that any trajectory that Johnson calculated would or could have not taken this time shift into account.
The point is that while it was possible to know how to calculate going up, and to know how to calculate an orbit, it was necessary to make a calculation of a combination of both in order to launch into orbit with a specific re-entry zone in mind.
This is where Johnson employed Euler's method and invented new mathematics in order to calculate trajectories for the initial NASA space missions.
But to say so - I've completely lost my train of thought as to why it may of been useful for your relativistic correction thread. I know why I'm interested, and if I remember (might have to read back on the thread for that), for what reason I thought it may have been relevant to you, I'll be back.
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Johnson was certainly aware of relativity, but I'm pretty sure she didn't have to use it. The Newtonian equations are more than adequate to get you to the moon and back, but they are diabolically difficult to solve. You have to resort to numerical approximations like Euler's method. Relativity would not make that any easier and the increase in precision would be miniscule.
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I think I know why the new metric is invalid. The derivation of relativistic escape velocity assumes coordinate light speed is invariant and that is only true in the absence of gravity. If you apply the variable speed of light (VSL) hypothesis to the old metric, the equations of motion for the radial free fall case are:
dr/dt=-c*(1-rs/r)*sqrt(rs/r)
dT/dt=1-rs/r
This solves the penetration paradox because everything (including light) stops at the horizon. It should also solve the dark matter dilemma because it predicts slower orbits near the horizon. Another interesting result is that PE=moc2 at the horizon. Conservation of energy in the extreme!
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Here's an illustration of this result. Notice the v/c profile. It's a good match to a galaxy rotation curve.
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On 2nd thought, I'm not sure if this actually invalidates the new metric. However, the derivation of relativistic escape velocity on which it is based (http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm) uses relativistic mass at escape velocity to calculate the Newtonian potential. That's a dubious step because the mass is not moving in that context. In that case:
PE=GMmo/r (rather than PE=GMmo*gamma/r)
KE=moc2*(gamma-1)
The SC metric is consistent with these relations (https://en.wikipedia.org/wiki/Kinetic_energy). Jeffrey was right to suspect the validity of that derivation (https://www.thenakedscientists.com/forum/index.php?topic=69595.msg506434#msg506434).
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I'm still intrigued by the VSL interpretation though.
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I don't see any way to notify the author about his mistake on his website. He seems to be quite prolific (and consuming public funds in the process.) Hopefully the peer review process will catch it. In the mean time, be wary of publications by Joseph A. Rybczyk.
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I don't see any way to notify the author about his mistake on his website. He seems to be quite prolific (and consuming public funds in the process.) Hopefully the peer review process will catch it. In the mean time, be wary of publications by Joseph A. Rybczyk.
There is little point in trying to contact him. Everyone in the trade is well aware of him and his ideas, but most have given up trying to argue with him. That's why you had a few shots across the bows in the original discussion then folks in the know stopped contributing when they saw where you were going. A lot of people have a limited amount of discussion time available so they cherry pick what they want to get involved in.
But you had fun getting there, yes?
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Absolutely. Thanks to all who contributed to my education in this matter.
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Rats. I thought I had it figured out, but if you correct for Rybczyk's error by removing gamma from the Newtonian potential then conservation of energy in the radial free fall case gives:
1-v2/c2=1/(1+rs/2r)
That leads to this metric:
c2dT=c2dt2/(1+rs/2r)2-(1+rs/2r)2dr2-r2d(angle)2
Like my original formulation, this is not a vacuum solution in strong fields. (i.e. The Ricci scalar is zero, but the Ricci tensor is not.) However, this one is:
c2dT=c2dt2/(1+rs/r)-(1+rs/r)[dr2-r2d(angle)2
I don't understand why this metric is any less plausible than the Schwarzschild one.
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The scaling factor for a vacuum solution evidently requires a classical approximation of KE, but there are two ways to skin that cat depending on whether you're trying to approximate KE or E-KE. The distinction is moot for weak fields, but one way leads to an event horizon and the other does not. How do you choose?
I think it all boils down to the fact that PE is a classical concept. The ratio rs/r is really PE/moc2. That is, the ratio of a classical quantity (PE) to a relativistic one (rest mass energy.) The form of the SC metric depends on how you define PE and that's a sticky wicket because PE is a property of the system, not the test mass. I understand that gauge theory avoids the issue by representing PE in terms of 4-momentum, but I don't think it can distinguish between the two possibilities. Higgs treats it as a scalar field so he should be happy either way. I'm stumped.
Maybe we need to account for the fact that the center of gravity is affected by the presence of the smaller mass. It's a negligible effect if they are sufficiently separated, but not when they get up close and personal. The event horizon may be an artifact of that error.
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I should clarify my point about classical approximations because some GR theorists take exception to the idea that the scaling distance in the SC solution emerges from Newtonian dynamics rather than the other way around. I don't have access to authoritative texts on the subject, but all of the derivations I've seen invoke Newtonian dynamics to define the scaling distance. In either case though, the ratio rs/r equates to v2/c2 for radial free fall and the question is then how to determine v in strong fields because all GR can tell you is how it depends on time dilation.
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I found a derivation that seems to bypass Newton (link below.) The idea is that curvature imposes a pressure that cancels Newtonian gravity so I think it still involves Newton in a roundabout way, What puzzles me is why this metric doesn’t crop up as a possible solution for the integral of Gtt since it satisfies all of the same constraints. He doesn’t show his work for that part so it’s going to take some ciphering to diagnose. I suspect there's a +/- missing in one of the Christoffel symbols.
http://www.google.ca/url?url=http://gfm.cii.fc.ul.pt/events/lecture_series/general_relativity/gfm-general_relativity-lecture4.pdf&rct=j&q=&esrc=s&sa=U&ved=0ahUKEwioyZ6h5JjTAhUnI8AKHf_aDjMQFggUMAA&usg=AFQjCNFEW6NdUweWkTaqo6fe9_zEF-i84Q
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Never mind. I mixed up the time and radial coordinates so I was solving for the case of negative mass. It's interesting that there's no horizon in that case, but I can't imagine any practical applications for that solution. My local expert (Wilson) is aware of a researcher in Montreal who has pondered such a thing, but it evidently didn't produce any new insights.
So once again, the new metric is debunked unless there is any significance to solutions that only approximate the vacuum.
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The only useful take away from all of this is the VSL interpretation. It's not a new idea, but it may be worth a second look because it does solve the penetration paradox. More on that in the original thread: https://www.thenakedscientists.com/forum/index.php?topic=69595.0
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Kevin Brown talks about the alternate metric here: http://mathpages.com/rr/s5-05/5-05.htm
By his account, it's a viable solution except that it predicts the wrong value for the precession of Mercury's orbit.