rjbeery

The reason it's difficult to discuss is because "constant g" is unnatural. It's mathematically impossible, so the premise is invalid. You're thinking in terms of "constant acceleration" as somehow adding to a value of velocity, which would increase the time dilation, but gravitational acceleration isn't true acceleration -- free falling objects are unaccelerated by definition. The bottom line is that there can't be stretches of space where velocity increases but gravitational acceleration remains constant. Potential is defined with r, g is defined with r, so potential cannot be independent of g.

No problem. Please provide me with an exact equation showing me this.

OK, so I've never looked at the derivation of the approximation before. This is a great synopsis: https://campus.mst.edu/physics/courses/409/Assignments/gravitational potential.pdf I think I understand. The approximation U = mgh has 'g' baked into it; we could rewrite this as g = U/mh, and if we take the derivative with respect to r (or h), the change in g is independent of a change in r (or h) in this form -- so you are right. However, the true math (sans approximation) is a full Taylor series expansion, with the height variable continuing on indefinitely to higher and higher powers. In other words, the full expression of g = U/mh is infinitely differentiable with respect to r and will always be dependent upon r. There is no scenario where the acceleration can exist in a form that is independent of r. So, yes, the approximation is exactly the cause of our disagreement.

I think it's important that we agree on where we disagree. I don't deny anything about Einstein's elevator, except that it isn't a true equivalence if the acceleration is due to gravity, and that's because I don't believe "constant acceleration due to gravity" is possible, and the Newtonian approximation is obfuscating that fact. Light would obviously bend under acceleration, regardless of the source. You're asking me to prove that your approximation is an approximation. I may look at doing this, but at this point I don't see much benefit in convincing you as long as you haven't found any objective refutation in the mathematics of my paper.

It doesn't give the same answer. Saying "the other [terms] are small and can be ignored" is like saying that pi is literally 333/106. If Einstein's elevator is your example of a physical scenario where potential increases over a substantial distance of constant gravitational acceleration, it isn't valid. It too is an approximation. We can imagine, in our minds, an elevator being accelerated uniformly from top-to-bottom, but the person standing in the elevator could use equipment (available today) to compare the gravitational acceleration at his head vs his feet. If they differ then he knows he is in a gravitational field. A gravitational field is defined as the negative of the gradient of the gravitational potential. If the gradient is zero, then the gravitational acceleration is zero -- how would you explain that?

But it isn't. I agree with you that time dilation, and therefore refraction, are correlated with gravitational potential, but a change in gravitational potential requires a change in time dilation. I have given the full GR treatment of velocity and how it relates to a free fall in gravity in equations (1)-(6). I do the same in the second section with refraction. Yes, I did this, and we both agree that my math is an exact result, whereas the Newtonian approximation used by Pound-Rebka is just that. It is known to be inexact, and would quickly diverge from reality as the height of the tower increased. In other words, you're using a known approximation as a proxy for GR when we both know it is not. If you still disagree then I would like to examine a physical scenario where potential increases over a substantial distance of constant gravitational acceleration. The only one I can think of is in the center of a Newton's sphere (where potential remains unchanged).

I understand now. You think that, because the approximation has varying time dilation with a constant g, then that's proof that GR claims the same thing. This is false, and equivalent to saying that Newton's approximations prove that the speed of light is infinite.

What different result were you referring to here? To be honest, the things you're saying are just bizarre. This is from the Pound-Rebka wiki page: In other words, treating g as constant is a known approximation, and the proper calculation is literally listed. I solved that exact calculation to show you that it does not produce a "different result". If we used exact values for r_s and altitude then it would produce an exact result.

I can't tell if you're being intentionally obtuse, but .009 is the approximate Schwarzschild radius of the Earth. I used that for r_s in the calculation. The above math was my response to you saying where the ratio of time dilation factors (using rough estimates) is a difference of 3*10^(-15), compared to the gh/c^2 = 2.5*10^(-15). The fact that they could not (at the time) measure the time dilation difference at a distance of 22.5 meters, forcing them to use the approximation you listed, does not mean that the GR answer is anything but correct.

Have you done any calculations? and gh/c^2 = 2.5×10^(−15)

This isn't true; the frequency shift you're using is a Newtonian approximation. Compare this (the true equation): to equation (12) in my paper above and you can see that they are clearly just taking the ratio of time dilation at both heights to determine blueshift.

There's nothing controversial in the paper. It's very straightforward. If you think this paper is inconsistent with GR then you are perhaps misrepresenting what GR predicts. I don't understand the second statement above. I'm not sure how (or why) I would want to derive time dilation without GR. Lastly, time dilation would exist in constant g but it would also be constant, and therefore gravitational acceleration would not be present. Consider the inside of a massive, transparent Newton's shell -- time moves slowly, but everything is free-floating.

According to this paper, "constant gravitational acceleration" would not induce forces at all. I mean, intuitively, we would expect it to (because we are accustomed to using g for convenience) but, as the paper outlines, time dilation would be constant in a field of "constant gravitational acceleration" and therefore would not refract light. If gravitational forces still existed in such a field then equivalence would be broken. Also, you are asking for the causal mechanism, but that is literally what the paper is about. Time dilation is the mechanism. If you're driving on a dirt road and you hit the shoulder of sand, your car turns in that direction; drop a straw in water (where light moves more slowly) and the straw appears to "bend down". It's all the same thing.

I copied your post above, but here's a direct link: https://www.scienceforums.net/topic/122084-paper-a-causal-mechanism-for-gravity/page/2/?tab=comments#comment-1141752 I'll be curious to see if others find this result unsurprising as well. I think it is very significant.

You asked for an explanation 11 months ago, and now you seem to be saying that it's obvious.

swansont, I feel that this paper is a completely different angle on a similar subject. Please consider letting it remain in its own thread. equation (19) should read:

In this paper, we show that the time dilation field of mass-energy is sufficient to predict gravitational effects on light. Essay written for the Gravity Research Foundation 2021 Awards for Essays on Gravitation. Free-falling into Black Hole We start with a black hole, B, possessing a Schwarzschild radius (rs) of 3000 m, giving it a mass of roughly one solar-mass (~2*10^30 kg). We take a body, A, of negligible mass, initially resting at a great distance (PEi = KEi = 0), and allow it to free-fall towards B. To calculate A’s coordinate velocity at a given distance, r, from the center of B, we start with (1) (2) (3) (4) (5) so (6) Integrating gives us: (7) We evaluate the equation for the final 1000 meters of A’s path before reaching the event horizon (i.e. r = 4000..3000, see Fig 2) (8) (9) So (10) Here we take note of the proper velocity of A at r = 4000 (see Fig 2) (11) We also note that (12) where t0 is the proper time of events for A, tf is the coordinate time of those same events (for a distant observer), and the radical value is the time dilation factor which approaches 0 as r approaches the event horizon at rs. Light passing through a graded refractive index A refractive index of a medium is defined as the dimensionless number (13) where v is the measured velocity of light through that medium. In other words, n can be treated as the reciprocal of an apparent time dilation factor (from the remote observer’s point of view). If we consider a gravitational field as the medium being traversed, then we can use (12) to represent that medium’s refractive index as (14) where ys is analogous to the Schwarzschild radius of B above. We see that as a light ray R approaches a height of ys the “time dilation factor” approaches zero, and n diverges to infinity. Light in this area is effectively frozen, and, for all intents and purposes, the horizontal boundary of y = ys is an event horizon. Now we take Snell’s Law (15) where k is a constant determined by the initial angle and location in the medium of an incident ray. Combining (14) with (15) we now have (16) so (17) In Figure 1 we are considering theta to be the angle between R and the normal to the x-axis, therefore (18) We can combine (16-18) to get (19) We now want to determine k. Since we know from (11) that the body A is approaching B at a velocity of .866025c at r = 4000 (see Fig 2), we choose theta such that the light ray R is approaching B at the same rate at y = 4000. A light ray with a vertical component moving downward at .866025c is doing so at pi/6 radians off the y-axis (20) (21) (22) All such k-values will be unity when the vertical component of R is equal to a radial free-fall velocity, such as A’s, at a given height. Plugging k = 1 into (19) we have (23) We now calculate the length of R’s spatial path from y = 4000 to the so-called event horizon ys = 3000. (24) (25) (26) (27) (28) So (29) which is the same value we calculated in (10). Gravity and refraction equivalence We can verify the relationship between (8) and (27) by adjusting (7) (30) (31) (32) where of course the constants of integration are irrelevant to the integral. Conclusion In conclusion, we have shown that the time dilation field of mass-energy predicts gravitational effects on light. Parsimony and equivalence would suggest that this mechanism is sufficient to explain gravitational forces on massive objects as well, possibly in the form of EM mass.

You're presuming that either the object in that image is a stable neutron star or a fully-formed black hole. A "frozen star" would still asymptotically red-shift light, it's just that the traditionally predicted effects of an event horizon would exist at r=0. This is one of those times that my BS-meter is going off. If the static solution gives a full accountability of the future then introducing a change in M does not change that fact. If I may be blunt, I believe you're just presuming that the Vaidya solution says something which bolsters your argument. You've even hedged your bet by saying You've thrown out what I believe to be a false resolution to my objection, and then you say This is intellectual insincerity. I raised an objection in the form of a logical contradiction in the OP. In an effort to respond to questions, I found three peer-reviewed papers which discuss my objection in detail (using the Schwarzschild metric), and agree with my objection. You throw out a comment about how the Vaidya solution resolves all problems, which you cannot prove but "may prove in the future", and until then we should just end the conversation. Is making a concession really that difficult for you?

Accretion disks are still expected from a very compact, high mass area.

The information paradox is resolved in this (and other) papers by claiming that the event horizon does not, ever, form, which is exactly what I've been saying. My objection to the original diagram used in the Hawking paper remains valid, and the idea that the Vaidya analysis solves this is false -- Vaidya black holes do not allow the infinite observer to see the event horizon in finite time, and that's the crux of the problem. If I were going to criticize anything about this and related papers, it's the fact that they are using Hawking "back reaction" to discuss the idea that the event horizon cannot form, but the rate of Hawking radiation is a function of the radius of the event horizon -- in other words, the black hole is a prerequisite for any back reaction of Hawking radiation to exist. This is why I'm careful to talk about "evaporating processes" without referring to Hawking radiation explicitly.

I also work under the presumption that there is a single reality. I have a real problem (again, philosophical, but I believe that it has merit) with Kruskal coordinates and their supposed resolution to the so-called mathematical singularity problem at the event horizon.

I understand the theory of studying "how {variable} changes with respect to {dimension 1} in the {dimension 2} direction". I can read Einstein tensor notation. I know the Schwarzschild metric pretty well. I understand Kruskal coordinates. This is neither here nor there, though, because I tend to analyze Physics on a more philosophical level. When the math "says something" I try very hard to understand what it's saying physically, and when I don't I try to find someone who does and ask them to explain it.

The paper I referred to has addressed my issues and acknowledged the contradictions I believed to exist in the conventional model. It was a relief to find, actually, being published in a prominent journal, Nuclear Physics B, in 2016. My frustrations were doubled up in this thread because some posters' responses implied that 1) the issues I raised were trivially explained, 2) the physics community was well-aware of them, 3) the responders personally understood the explanations but, 4) the math was too complex to explain it to laymen. My BS-meter wouldn't stop blinking. Anyway, thanks for the back-and-forth on this.

Agreed, which means that either 1) General Relativity is wrong, 2) some mysterious delimiter on the velocity of cosmic rays prevents them from reaching the critical energy required for a predicted micro black hole, or 3) the micro black holes evaporate before they can be detected. I'm working under the presumption of #3, and I feel it's fair to say that this is the same position held by most of the physics community today (as evidenced by the paper I linked to regarding the Large Hadron Safety Assessment Group). This is close, but regarding #1 -- the interior of a black hole cannot be described by the remote observer, and does not need to be. The infinite future is represented for a coordinate time observer B before mass crosses the event horizon. We have to take this literally. Pick a method of determining simultaneity, and then map those events of object A approaching the event horizon to the remote observer B; there will be events A that match to events B for any and all times/events for B from "now" until "eternity". Now, let the black hole evaporate, such that the black hole and A are simply gone and replaced by a new observer, C. Now, B and C can match events using our method of simultaneity, forever. In other words, If B were to describe what was happening in the region of the black hole at some certain time, he would claim that both A is asymptotically approaching it, and C is calmly residing there with no black hole in sight. I found this paper today, and, based on the abstract, it's making a similar argument: Parse this paragraph carefully. It says that it's assumed that an event horizon can form with Hawking radiation...but we have no model for it. Therefore we "turn off" Hawking radiation to allow the event horizon to form, and then "turn it on" to produce the graph that I used to show that a contradiction resides in this logic.

Of course, and if the physical conditions were right then those things may come to be...or not. But the physical conditions for micro black holes are likely met continuously in our atmosphere. Obviously c applies to cosmic rays - when I said "no known mechanism might limit their incoming velocity" I meant limit it as a percentage of c such that they would be prevented from having sufficient energy to predict a micro black hole. Your second comment is baffling. Which assumption is false? Micro black holes do not produce cosmic rays -- they are produced by them. If you could summarize MY position on this, how would you do so?