QM + GR = black holes cannot exist
- Thread starter RJBeery
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Thanks Tashja. I'll go through it. It's interesting.
This is a good example of how intuitive models can go astray. The argument presumes that the light signal does not "slow down,"
No it doesn't. I know full well that that light "slows down". The scenario is intended to tease it out, and the implications. IMHO this leads you to the conclusion that the Schwarzschild singularity at the event horizon isn't just a coordinate artefact, and that Oppenheimer's original "frozen star" black hole interpretation is the one that's correct.
but what exactly does that mean? An observer at rest relative to the star will always measure the outgoing light signal to have speed c *locally,* (that is, as the flash passes through a laboratory that is very small compared to scale over which spacetime is locally curved), but to talk about the speed of a signal emerging from the planet's surface and going all the way to infinity, one needs a *global* coordinate system (one that applies at all positions in spacetime, such as the Schwarzschild coordinate system) to talk about the signal's speed at various points. An observer using such a coordinate system will find that the light flash will move *slower* than c close to the planet's surface than it does at at infinity. This does not contradict the previous results, because time runs more slowly for observers close to the planet's surface than for those higher up, so what looks like something moving with speed c to an observer close to the surface looks like something moving slower to someone whose clock is running faster.
All good stuff. We are talking about the "coordinate" speed of light here. Personally I think it's better to think of this as the speed of light, but no matter.
As the planet's mass approaches the black hole limit, the signal emitted from the surface will seem to move more and more slowly away from the surface (and will also be seen to be increasingly red-shifted as observed from infinity). When the surface of the planet coincides with the black hole's event horizon, the signal will stop moving outward from the surface (and the redshift observed at infinity will go to infinity). So light no longer escapes.
Very Good. Professor Moore passes the test with flying colours. The light doesn't get out because it's stopped. To be honest I'm rather surprised that he got it right. But wait a minute...
This also does not contradict the statement about an observer at rest on the surface seeing the signal to have speed c, because as event horizon moves beyond the planet's surface, that surface can no longer remain at rest, but in fact must go to r = 0 in a finite time (as measured by an observer on the surface)
Whoa whoa whoa. The force of gravity at some location relates to the local gradient in the "coordinate" speed of light. But at the event horizon, this is zero. And there's no such thing as a negative speed, so there's no more gradient, and so no more force of gravity. In addition light has stopped. That observer on the surface isn't measuring anything. His light clock has stopped, and he has stopped too. And he can't move faster than light, so he isn't going anywhere. His finite proper time is a mathematical myth in a neverneverland beyond the end of time. It hasn't happened yet, and it never ever will.
just as surely as the past must go towards the future.
The past doesn't go towards the future in any real sense. That's just a figure of speech. Time travel is science fiction, see this thread. A clock "clocks up" some kind of regular cyclical internal motion and displays a cumulative result called "the time". When a clock goes slower it's because that motion goes slower. And when a light clock has stopped because the light has stopped, that clock isn't going anywhere.
Even then, an observer on the surface will *still* see the light moving outward at speed c, but from the perspective of the global coordinate system, it is simply that the observer is falling faster toward r = 0 than the signal is.
No, the observer doesn't see anything. Light has stopped, and so have electrochemical signals in his brain.
To understand all this fully, I strongly recommend that the questioner take a course in general relativity!
I think I do understand it fully, because I've read the original GR material. I've also read about Friedwardt Winterberg's GRB firewall, but that's one for another day.
A predictable Farsight answer: take the part he likes, deny the rest (if he acknowledges it at all).
Farsight, here is another example of someone pointing out that your limited interpretation only fits one specific class of reference frames, not GR in general. You want to stick to a notion of absolute space and time that doesn't match GR in general.
Farsight, if you think that Prof. Moore is wrong about what happens to the observer on the surface, work out the system of coordinates associated with that observer. For an expert on GR like yourself, that should be no problem.
Farsight, here is another example of someone pointing out that your limited interpretation only fits one specific class of reference frames, not GR in general. You want to stick to a notion of absolute space and time that doesn't match GR in general.
Farsight, if you think that Prof. Moore is wrong about what happens to the observer on the surface, work out the system of coordinates associated with that observer. For an expert on GR like yourself, that should be no problem.
RajeshTrivedi
Valued Senior Member
Plain speaking at event horizon time stops (Rc = r) and consequences of taking this statement literally will give obnoxious interpretations, thats what farsight is doing.
In the farsight thought experiment he is practically standing on the surface of a star collapsing towards becoming the black hole. This process will be completed in finite time, only problem is that when he passes through event horizon then Schwarzschild coordinate becomes infinity, but that does not relate with anybody's proper time. It is just the coordinate. The time t corresponds to future direction outside the event horizon, but inside the even horizon time is actually spatial.
hilarious
It isn't something I've dreamt up, Rajesh. Kevin Brown talks about two different interpretations in The Formation and Growth of Black Holes:
"Historically the two most common conceptual models for general relativity have been the "geometric interpretation" (as originally conceived by Einstein) and the "field interpretation" (patterned after the quantum field theories of the other fundamental interactions). These two views are operationally equivalent outside event horizons, but they tend to lead to different conceptions of the limit of gravitational collapse. According to the field interpretation, a clock runs increasingly slowly as it approaches the event horizon (due to the strength of the field), and the natural "limit" of this process is that the clock asymptotically approaches "full stop" (i.e., running at a rate of zero). It continues to exist for the rest of time, but it's "frozen" due to the strength of the gravitational field. Within this conceptual framework there's nothing more to be said about the clock's existence. In contrast, according to the geometric interpretation, all clocks run at the same rate, measuring out real distances along worldlines in curved spacetime. This leads us to think that, rather than slowing down as it approaches the event horizon, the clock is following a shorter and shorter path to the future time coordinates..."
He doesn't favour the "field interpretation", but I think it's the one that's right. Note that this article used to refer to Wheeler and Weinberg instead of Einstein and quantum field theory. Kevin Brown changed it after I queried it with Weinberg. Only Einstein would never have agreed with the idea of a black hole with a point singularity in the middle.
That's what people tend to say, but I'm confident that it's the wrong interpretation, and that if Einstein was around, he'd back me up. IMHO it all comes back having the correct understanding of time, see time travel is science fiction and pay special attention to A World Without Time: The Forgotten Legacy of Godel and Einstein.
Bullshit. Utter crap!
Light is never seen to be stopped in any FoR.
From a remote FoR, light is slowed and red shifted to infinity....
From the local FoR, light proceeds at "c" from that observers point of view, time proceeds as normal within that local frame, and nothing strange happens. [ignoring tidal gravitational effects as one crosses the EH]
Now Farsight from one layman to another obvious layman, that is the accepted mainstream model as to what happens, and highlights one of the postulates of relativity that all FoR's are as valid as each other.
As the good professor has told you, you are wrong.
Obviously he has something to say, and as erroneous and incorrect as that is, his ego insists that he says it, and as public forums such as this are his only outlet, we consequently are the vehicle for this diatribe.
It appears GR gravity does have limitations in that although it can dilate time, it has no effect on such delusions of grandeur driven by that ego.
The following is a reasonable answer to a FAQ from.....
http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/fall_in.html
What happens to you if you fall into a black hole?
Suppose that, possessing a proper spacecraft and a self-destructive urge, I decide to go black-hole jumping and head for an uncharged, nonrotating ("Schwarzschild") black hole. In this and other kinds of hole, I won't, before I fall in, be able to see anything within the event horizon. But there's nothing locally special about the event horizon; when I get there it won't seem like a particularly unusual place, except that I will see strange optical distortions of the sky around me from all the bending of light that goes on. But as soon as I fall through, I'm doomed. No bungee will help me, since bungees can't keep Sunday from turning into Monday. I have to hit the singularity eventually, and before I get there there will be enormous tidal forces—forces due to the curvature of spacetime—which will squash me and my spaceship in some directions and stretch them in another until I look like a piece of spaghetti. At the singularity all of present physics is mute as to what will happen, but I won't care. I'll be dead.
For ordinary black holes of a few solar masses, there are actually large tidal forces well outside the event horizon, so I probably wouldn't even make it into the hole alive and unstretched. For a black hole of 8 solar masses, for instance, the value of r at which tides become fatal is about 400 km, and the Schwarzschild radius is just 24 km. But tidal stresses are proportional to M/r3. Therefore the fatal r goes as the cube root of the mass, whereas the Schwarzschild radius of the black hole is proportional to the mass. So for black holes larger than about 1000 solar masses I could probably fall in alive, and for still larger ones I might not even notice the tidal forces until I'm through the horizon and doomed.
Won't it take forever for you to fall in? Won't it take forever for the black hole to even form?
Not in any useful sense. The time I experience before I hit the event horizon, and even until I hit the singularity—the "proper time" calculated by using Schwarzschild's metric on my worldline—is finite. The same goes for the collapsing star; if I somehow stood on the surface of the star as it became a black hole, I would experience the star's demise in a finite time.
On my worldline as I fall into the black hole, it turns out that the Schwarzschild coordinate called t goes to infinity when I go through the event horizon. That doesn't correspond to anyone's proper time, though; it's just a coordinate called t. In fact, inside the event horizon, t is actually a spatial direction, and the future corresponds instead to decreasing r. It's only outside the black hole that t even points in a direction of increasing time. In any case, this doesn't indicate that I take forever to fall in, since the proper time involved is actually finite.
At large distances t does approach the proper time of someone who is at rest with respect to the black hole. But there isn't any non-arbitrary sense in which you can call t at smaller r values "the proper time of a distant observer," since in general relativity there is no coordinate-independent way to say that two distant events are happening "at the same time." The proper time of any observer is only defined locally.
A more physical sense in which it might be said that things take forever to fall in is provided by looking at the paths of emerging light rays. The event horizon is what, in relativity parlance, is called a "lightlike surface"; light rays can remain there. For an ideal Schwarzschild hole (which I am considering in this paragraph) the horizon lasts forever, so the light can stay there without escaping. (If you wonder how this is reconciled with the fact that light has to travel at the constant speed c—well, the horizon is traveling at c! Relative speeds in GR are also only unambiguously defined locally, and if you're at the event horizon you are necessarily falling in; it comes at you at the speed of light.) Light beams aimed directly outward from just outside the horizon don't escape to large distances until late values of t. For someone at a large distance from the black hole and approximately at rest with respect to it, the coordinate t does correspond well to proper time.
So if you, watching from a safe distance, attempt to witness my fall into the hole, you'll see me fall more and more slowly as the light delay increases. You'll never see me actually get to the event horizon. My watch, to you, will tick more and more slowly, but will never reach the time that I see as I fall into the black hole. Notice that this is really an optical effect caused by the paths of the light rays.
This is also true for the dying star itself. If you attempt to witness the black hole's formation, you'll see the star collapse more and more slowly, never precisely reaching the Schwarzschild radius.
Now, this led early on to an image of a black hole as a strange sort of suspended-animation object, a "frozen star" with immobilized falling debris and gedankenexperiment astronauts hanging above it in eternally slowing precipitation. This is, however, not what you'd see. The reason is that as things get closer to the event horizon, they also get dimmer. Light from them is redshifted and dimmed, and if one considers that light is actually made up of discrete photons, the time of escape of the last photon is actually finite, and not very large. So things would wink out as they got close, including the dying star, and the name "black hole" is justified.
As an example, take the eight-solar-mass black hole I mentioned before. If you start timing from the moment the you see the object half a Schwarzschild radius away from the event horizon, the light will dim exponentially from that point on with a characteristic time of about 0.2 milliseconds, and the time of the last photon is about a hundredth of a second later. The times scale proportionally to the mass of the black hole. If I jump into a black hole, I don't remain visible for long.
Also, if I jump in, I won't hit the surface of the "frozen star." It goes through the event horizon at another point in spacetime from where/when I do.
(Some have pointed out that I really go through the event horizon a little earlier than a naive calculation would imply. The reason is that my addition to the black hole increases its mass, and therefore moves the event horizon out around me at finite Schwarzschild t coordinate. This really doesn't change the situation with regard to whether an external observer sees me go through, since the event horizon is still lightlike; light emitted at the event horizon or within it will never escape to large distances, and light emitted just outside it will take a long time to get to an observer, timed, say, from when the observer saw me pass the point half a Schwarzschild radius outside the hole.)
All this is not to imply that the black hole can't also be used for temporal tricks much like the "twin paradox" mentioned elsewhere in this FAQ. Suppose that I don't fall into the black hole—instead, I stop and wait at a constant r value just outside the event horizon, burning tremendous amounts of rocket fuel and somehow withstanding the huge gravitational force that would result. If I then return home, I'll have aged less than you. In this case, general relativity can say something about the difference in proper time experienced by the two of us, because our ages can be compared locally at the start and end of the journey.
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Obviously the answers not only refutes "Farsight Physics" on BH's and gravity, it also refutes his often made claim that time travel [although still sci/fi at this time] is Impossible.
No physical law or GR disallows time travel.
And many practical examples of how it could occur are available.
Also probably the world's foremost authority on BH's Kip Thorne, is someone who Farsight needs to take notice of.
Some good books I recommend to him that may help him see the light, and reality as it really is.
BLACK HOLES and TIME WARPS:
Kip Thorne:
GRAVITY'S FATAL ATTRACTION:
Sir Martin Rees and Mitch Begalman:
GRAVITATION:
Misner, Thorne Wheeler:
http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/fall_in.html
What happens to you if you fall into a black hole?
Suppose that, possessing a proper spacecraft and a self-destructive urge, I decide to go black-hole jumping and head for an uncharged, nonrotating ("Schwarzschild") black hole. In this and other kinds of hole, I won't, before I fall in, be able to see anything within the event horizon. But there's nothing locally special about the event horizon; when I get there it won't seem like a particularly unusual place, except that I will see strange optical distortions of the sky around me from all the bending of light that goes on. But as soon as I fall through, I'm doomed. No bungee will help me, since bungees can't keep Sunday from turning into Monday. I have to hit the singularity eventually, and before I get there there will be enormous tidal forces—forces due to the curvature of spacetime—which will squash me and my spaceship in some directions and stretch them in another until I look like a piece of spaghetti. At the singularity all of present physics is mute as to what will happen, but I won't care. I'll be dead.
For ordinary black holes of a few solar masses, there are actually large tidal forces well outside the event horizon, so I probably wouldn't even make it into the hole alive and unstretched. For a black hole of 8 solar masses, for instance, the value of r at which tides become fatal is about 400 km, and the Schwarzschild radius is just 24 km. But tidal stresses are proportional to M/r3. Therefore the fatal r goes as the cube root of the mass, whereas the Schwarzschild radius of the black hole is proportional to the mass. So for black holes larger than about 1000 solar masses I could probably fall in alive, and for still larger ones I might not even notice the tidal forces until I'm through the horizon and doomed.
Won't it take forever for you to fall in? Won't it take forever for the black hole to even form?
Not in any useful sense. The time I experience before I hit the event horizon, and even until I hit the singularity—the "proper time" calculated by using Schwarzschild's metric on my worldline—is finite. The same goes for the collapsing star; if I somehow stood on the surface of the star as it became a black hole, I would experience the star's demise in a finite time.
On my worldline as I fall into the black hole, it turns out that the Schwarzschild coordinate called t goes to infinity when I go through the event horizon. That doesn't correspond to anyone's proper time, though; it's just a coordinate called t. In fact, inside the event horizon, t is actually a spatial direction, and the future corresponds instead to decreasing r. It's only outside the black hole that t even points in a direction of increasing time. In any case, this doesn't indicate that I take forever to fall in, since the proper time involved is actually finite.
At large distances t does approach the proper time of someone who is at rest with respect to the black hole. But there isn't any non-arbitrary sense in which you can call t at smaller r values "the proper time of a distant observer," since in general relativity there is no coordinate-independent way to say that two distant events are happening "at the same time." The proper time of any observer is only defined locally.
A more physical sense in which it might be said that things take forever to fall in is provided by looking at the paths of emerging light rays. The event horizon is what, in relativity parlance, is called a "lightlike surface"; light rays can remain there. For an ideal Schwarzschild hole (which I am considering in this paragraph) the horizon lasts forever, so the light can stay there without escaping. (If you wonder how this is reconciled with the fact that light has to travel at the constant speed c—well, the horizon is traveling at c! Relative speeds in GR are also only unambiguously defined locally, and if you're at the event horizon you are necessarily falling in; it comes at you at the speed of light.) Light beams aimed directly outward from just outside the horizon don't escape to large distances until late values of t. For someone at a large distance from the black hole and approximately at rest with respect to it, the coordinate t does correspond well to proper time.
So if you, watching from a safe distance, attempt to witness my fall into the hole, you'll see me fall more and more slowly as the light delay increases. You'll never see me actually get to the event horizon. My watch, to you, will tick more and more slowly, but will never reach the time that I see as I fall into the black hole. Notice that this is really an optical effect caused by the paths of the light rays.
This is also true for the dying star itself. If you attempt to witness the black hole's formation, you'll see the star collapse more and more slowly, never precisely reaching the Schwarzschild radius.
Now, this led early on to an image of a black hole as a strange sort of suspended-animation object, a "frozen star" with immobilized falling debris and gedankenexperiment astronauts hanging above it in eternally slowing precipitation. This is, however, not what you'd see. The reason is that as things get closer to the event horizon, they also get dimmer. Light from them is redshifted and dimmed, and if one considers that light is actually made up of discrete photons, the time of escape of the last photon is actually finite, and not very large. So things would wink out as they got close, including the dying star, and the name "black hole" is justified.
As an example, take the eight-solar-mass black hole I mentioned before. If you start timing from the moment the you see the object half a Schwarzschild radius away from the event horizon, the light will dim exponentially from that point on with a characteristic time of about 0.2 milliseconds, and the time of the last photon is about a hundredth of a second later. The times scale proportionally to the mass of the black hole. If I jump into a black hole, I don't remain visible for long.
Also, if I jump in, I won't hit the surface of the "frozen star." It goes through the event horizon at another point in spacetime from where/when I do.
(Some have pointed out that I really go through the event horizon a little earlier than a naive calculation would imply. The reason is that my addition to the black hole increases its mass, and therefore moves the event horizon out around me at finite Schwarzschild t coordinate. This really doesn't change the situation with regard to whether an external observer sees me go through, since the event horizon is still lightlike; light emitted at the event horizon or within it will never escape to large distances, and light emitted just outside it will take a long time to get to an observer, timed, say, from when the observer saw me pass the point half a Schwarzschild radius outside the hole.)
All this is not to imply that the black hole can't also be used for temporal tricks much like the "twin paradox" mentioned elsewhere in this FAQ. Suppose that I don't fall into the black hole—instead, I stop and wait at a constant r value just outside the event horizon, burning tremendous amounts of rocket fuel and somehow withstanding the huge gravitational force that would result. If I then return home, I'll have aged less than you. In this case, general relativity can say something about the difference in proper time experienced by the two of us, because our ages can be compared locally at the start and end of the journey.
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Obviously the answers not only refutes "Farsight Physics" on BH's and gravity, it also refutes his often made claim that time travel [although still sci/fi at this time] is Impossible.
No physical law or GR disallows time travel.
And many practical examples of how it could occur are available.
Also probably the world's foremost authority on BH's Kip Thorne, is someone who Farsight needs to take notice of.
Some good books I recommend to him that may help him see the light, and reality as it really is.
BLACK HOLES and TIME WARPS:
Kip Thorne:
GRAVITY'S FATAL ATTRACTION:
Sir Martin Rees and Mitch Begalman:
GRAVITATION:
Misner, Thorne Wheeler:
I don't really remember anyone really asking for your opinion about anything, Dr. Troll.
It isn't. And this "frozen star" black hole isn't something I've made up. Don't react with hostile emotion to something that's new to you.
Duh! You can't see that light is stopped if light is stopped.
Like Prof Moore said, the light doesn't get out of the black hole because it's stopped.
No. Nothing happens. It hasn't happened yet, and it never ever will.
It's just one interpretation, and IMHO it's wrong. I think the 1995 Matt McIrvin Baez article is wrong too. You really need to get use to the idea that some of the things you've read about might be wrong. Don't dismiss the possibility out of hand, examine the evidence, study the logic, think it through for yourself.
I'm not wrong about why the light doesn't get out. He agreed with me on that.
Note that I don't pay much attention to Kip Thorne because he believes in time travel, or Martin Rees because he's a multiverse guy. Also note that Wheeler was criticised in A World Without Time for ignoring and corrupting Einstein's work. And see Eddington-Finkelstein coordinates which feature in MTW even though Eddington and Finkelstein didn't devise them.
[1]Yes it's an Interpretation......The mainstream Interpretation which gives it some legitamcy having undergone peer review......
[2] Yes it is your opinion it is wrong.......That's OK, your opinion does not really count for that much in the greater scheme of things anyway.
[3]The only thing I'm dismissing out of hand is the delusions some people are under.
Well so far I have been witness to you, [1] taking great people out of context, [2] Misinterpreting expert opinion, [3] Misquoting references to align with your own Interpretation, and now [4] Ignoring probably the world's foremost expert on BH's and gravity as well as other mainstream professionals.
As I said, your opinion does not really count for much, considering other matters that you have claimed.
It's one of two interpretations. You didn't even know there were two. You think your "mainstream" explanation is the only explanation. Well it isn't. And since you know that the local force of gravity relates to the local gradient in the coordinate speed of light and light can't go slower than stopped, you can't justify it any gravity beyond the event horizon. Or are you still clinging to the cargo-cult nonsense wherein space falls into a black hole at the speed of light?
I haven't taken anybody out of context or misinterpeted or misquoted anybody. And I certainly don't ignore the experts, I'm forever giving references to papers and articles etc. I know I said I don't pay much attention to Kip Thorne, but like I've said before, you have to understand that time travel is science fiction to understand gravity. And he doesn't. See this.
By the by, your post quotes Layman instead of quoting me.
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You obviously have. In the last two days, you attributed a quotation to Einstein that does not appear in the piece you were citing. In that piece, Einstein made an explicit statement contradicting the absolute space & time interpretation of GR that you attribute to him.
Whether through lying or incompetence, you clearly misquoted Einstein here and in many other places.
Yes, articles that you clearly do not read or understand. For example, you cited an article as an argument against the idea that gravity was the result of spacetime curvature when that same article said, repeatedly, that the cause of gravity was spacetime curvature.
Again, through incompetence or dishonesty, you ignore the experts.
And not bothering to learn the actual science of Einstein, your most holy of experts, is clearly ignoring the experts.
So now you are either denying the twin scenario or you are denying that physicists should practice physics.
The most interesting quotation from that Thorne article is this, "These mechanisms (1) and (2) are descriptive translations of mathematical results that we physicists have derived using the laws of physics expressed in their own natural language: mathematics. The sentences (1) and (2) capture the essence of our calculations, but crucial details are lost in translation."
Farsight, you and I both know that you cannot do the mathematics. I don't even think that you have tried to learn the relevant mathematics. But without you actually learning the relevant mathematics and the physics that makes use of it, you can't be said to be meaningfully engaged in much of the physics you try to discuss. Thorne is presenting, in brief, the results of a lot of work through the physics, work you refuse to do because you stick dogmatically to the little axioms that you have adopted.
Please, learn physics. You might be very happy.
I've learned the physics. Hence I can tell you that space doesn't fall down in a gravitational field, and that time dilation as per the twins "paradox" is not time travel. And much else. As for mathematics, you and I know I can do it, and as for quoting Einstein, you and I both know I've quoted him faithfully from this. He said "It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds". So he thought of a gravitational field as a state of space. By the by, in this Baez article you can read this:
"Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial. "
Space isn't curved where a gravitational field is. Nor is it falling down. Instead it's inhomogeneous. As I've said previously, you can read Einstein saying this in his 1920 Leyden Address:
"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gmn), has, I think, finally disposed of the view that space is physically empty."
Paddoboy: do note that Einstein distinguishes between space and space-time. Also note that when you plot light-clock rates in an equatorial slice through the inhomogeneous space around the Earth, what you're essentially plotting is curved spacetime. Like the Riemann curvature depiction below. Where the clocks run slower, the plot is lower. It's really simple once the penny drops.
"Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial. "
Space isn't curved where a gravitational field is. Nor is it falling down. Instead it's inhomogeneous. As I've said previously, you can read Einstein saying this in his 1920 Leyden Address:
"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gmn), has, I think, finally disposed of the view that space is physically empty."
Paddoboy: do note that Einstein distinguishes between space and space-time. Also note that when you plot light-clock rates in an equatorial slice through the inhomogeneous space around the Earth, what you're essentially plotting is curved spacetime. Like the Riemann curvature depiction below. Where the clocks run slower, the plot is lower. It's really simple once the penny drops.
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