Tomorrow I'll try to make an analogy between local and remote measurements in the weak field [our solar system or very far away from the black hole] and local and remote measurements in the strong field [very near the black hole]. Both fields are the 'playground' for modern gravitational physics. I'll do my best to make sure the analogy works without a bunch of math.
Spacetime Is a Fairytale
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Tomorrow I'll try to make an analogy between local and remote measurements in the weak field [our solar system or very far away from the black hole] and local and remote measurements in the strong field [very near the black hole]. Both fields are the 'playground' for modern gravitational physics. I'll do my best to make sure the analogy works without a bunch of math.
This is often called the bowling ball on the rubber sheet. It's flawed in that it uses gravity, which pulls the bowling ball down, to give you a mental picture of the curved spacetime of a a gravitational field. A better analogy is to remove the ball and start with a flat rubber sheet. Then you mentally put your fingers together in the centre of the rubber sheet, and spread them. Then you replace the marble rolling across the rubber sheet with a ripple in the rubber sheet. To go one better you replace the rubber sheet with a 3-dimensional block of clear silicone rubber marked with lattice lines, and inject more silicone rubber in the centre where the Earth is. You then end up with a picture like this but with the lattice lines bulging out rather than in.
Excellent picture, thank you.
My pleasure Neverfly. I should point out that one still has to be a little cautious about this sort of thing. It's better than the bowling-ball analogy, but unless the Earth is a blue-streak worldline, it's a depiction of the gravitational field around the Earth, not a picture of "curved spacetime around the Earth". For the latter you can drop a dimension and take a horizontal slice through the picture (albeit with the lattice bulging outwards) at the Earth's equator, then add the time dimension to form a block. The block would be bulging outwards on all four vertical sides. Then if you drew the worldline of a lightbeam slanting up through the block, you'd see it to be curved. If however you draw a curved light-beam in the picture with the Earth in it, you're drawing a geodesic in space rather than spacetime.
Of course, if you really want to understand GR you need to do the maths. There's no getting around that.
One can understand concepts, just fine. Concepts are not a true understanding, but a general idea that satisfies most. But to understand it, math is required.
It's a bit of a hitch when asking about parts of Relativity where one has not understood the concept and none of the math and then they question the validity of Relativity.
You make a personally disparaging statement but do not explain just where and how I was being "intellectually dishonest" when I simply asked the question in context provided. Please justify your 'personal' attitude in the context of our present limited exchange, else you are merely bringing off-topic personal opinions into this thread. Thanks.
This is Chapter 2, Curving, from Taylor and Wheeler's text Exploring Black Holes. I'm going to ask you to look at several Figures.
http://www.eftaylor.com/pub/chapter2.pdf
Figure 2 on page 2-5 The significance: Over each path through curved spacetime a local segment of the path very closely approximates the flat spacetime of Special Relativity. Subsequently the effects of gravity can generally be ignored when making local empirical measurements [accounting for local miniscule gravitational effects won't change empirical measurements in a meaningful way]. One experiment where local gravitational effects must be accounted for is the GPS. Another experiment is the Gravity Probe B. The length of the path segment, where the effects of gravity can generally be ignored, is much larger in the weak field than a path segment in the strong field. For our solar system it's approximately 1 AU. Around the black hole it's very small. It's the reason why gravitational effects are ignored in the laboratory frame of the LHC. It means that we can generally use the simpler mathematics of SR for analyzing the local spacetime [physics].
Figure 6 and 7 on page 2-26 the dreaded embedding diagrams. They're actually very useful if you know what they signify and not just tossed out there for folks to wonder what they mean. We're going to use Figure 6 to understand where measurements are made for our comparative analogy between the local measurements and remote bookkeeper measurements of the same event in spacetime. The event is measuring the velocity of a stone falling towards the black hole. The remote velocity measurement is going to be dr/dt_bkkpr and the local velocity measurement is dr_shell/dt_shell. The local measurement is made in the proper frame of the falling stone as it crosses r_shell and the bkkpr measurement is made from remote coordinates far away.
2 formulas derived from GR
dr/dt_bkkpr = (1-2M/r)(2M/r)^1/2 [measurement made from the remote bkkpr coordinates]
dr_shell/dt_shell = (2M/r)^1/2 [local proper frame measurement made at r_shell as the stone passes over]
It's very easy to make the calculations by setting r_shell = nM. Example r=2M is the distance from the center of the black hole to the coordinate singularity at the event horizon. So r=4M is twice the distance and r=200M is 100 times the distance, etc.. So for making our comparisons in the weak and strong fields we can pick any r=nM that we choose. Large r=nM for the weak field and small r=nM for the strong field. We just substitute our choice for r=nM for r in the formulas.
I'll pick 2M/4M for the strong field and r=2M/20,000M for the weak field
Weak field
dr/dt_bkkpr = (1-2M/r)(2M/r)^1/2 = (1-2M/20,000M)(2M/20,000M)^1/2 = (.9999)(.01) = .00999c
And
dr_shell/dt_shell = (2M/r)^1/2 = (2M/20,000M)^1/2 = .01c
Remote and local measurements are very close.
Strong field
dr/dt_bkkpr = (1-2M/r)(2M/r)^1/2 = (1-2M/4M)(2M/4M)^1/2 = (.5)(.7071068) = .3535534c
dr_shell/dt_shell = (2M/r)^1/2 = (2M/4M)^1/2 = (.5)^1/2 = .7071068c
The remote measurement is 1/2 the local measurement.
The analogy becomes: the local measurement only accounts for spacetime curvature in the local proper frame where the stone velocity is measured at. The small area of the pumpkin. The remote measurement accounts for the entire spacetime curvature over the stones path. Over a long line on the pumpkin. That's why GR names the remote observer the Schwarzschild bookkeeper. Later in the chapter they discuss the model for bookkeeper measurements which are essentially global in nature.
Finally the end. Hope it works.
One final thing substitute rM=2 (2M for r at the event horizon) to get the greatest delta of all. Think about why the remote bookkeeper can no longer access information about the stone's future path inside the black hole. Switching to a proper frame metric where all the measurements are made by the stone rider allows GR to evaluate the spacetime inside the black hole.
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Thanks Farsight. That one's quite an improvement to the 2D one. Will think about it for a while.
Thanks brucep. I'll download this and the attached figures. I'll need a good deal of space / time to plough into it all, and of course, I'll keep my eyes open for what others here might say too.
It might look, at first glance, like it's something that you'll have to plow through but it's really not. Figure 2 is key to understand. Knowing they're are two different measurements made, one from a remote frame of reference and one from a local frame of reference, for comparison in each of field, weak and strong. Weak=the gravity in our solar system for example and strong=gravity around neutron stars and black holes. The analogy is the difference in the remote measurement and the local measurement. The local measurement only accounts for spacetime curvature where the local measurement is made while the remote measurement accounts for spacetime curvature over the entire path of the stone. The starting point of the falling stone is at rest far away. It's more than an analogy because it describes the real physics using spacetime curvature. The visual aid in Figure 2 is way better than the other stuff you've been looking at. Have a great day. Bruce
LaurieAG
Registered Senior Member
I think Prometheus was on a lunch break when he wrote his post. The SD is an attempt to build a equivalent model to GR from different parameters. I think the 'fairytale' comments are lame. In the original paper, from 2000, they open with it as if they need to justify building the model by noting the geometric theory works but is probably based on 'unphysical' parameters. I have a different opinion that measuring the geodetic effect with GP-B is empirical confirmation of spacetime geometry. Just my opinion.
LaurieAG
Registered Senior Member
I thought the paper brought back some balance that has been missing for a while. Have you ever read 'Through the Looking Glass' or 'Alice in Wonderland'? They were written by Lewis Carroll, a pen name for Charles Dodgson the Lucasian professor of Mathematics at Oxford.
In my opinion he was warning about the differences between the quantum world and the real world and getting trapped, like Alice, when you fail to find the way out, mathematically that is.
That's really interesting. I wasn't aware of that. I'm not against equivalent models. It would be good if they're successful. I still think the 'fairytale' comment is lame. Why bother? Crank fodder.
I find this statement to be very disturbing. Newtons Laws are not reconcilable with quantum mechanics, but does that mean that Newtons Laws are a fairy tale? I don't think it does. It only means that the quantum world is so much different than the marco world that the same laws cannot describe it accurately. It is also said that Einsteins description of nature are more accurate than Newtons Laws and Newtons Laws of motion are not completely correct, but then if you look at the beginning of Einsteins 1905 paper(that gave birth to modern physics) on the behavior of electrodynamics, it starts out using one of Newtons most basic equations. v=d/t So in reality what Einstein did was instead of saying that Newtons Laws of motion where wrong, he applied them to particles that travel the speed of light instead and "made the statement" that location of the law held priority in making a more accurate description of everything else, and then applying this basic law of motion to things like light had the consequence of altering how Newtons Laws describe other things in the world that does not behave like light.
Quantum Mechanics has become a game of fill in the blanks, we can only see what happens and try to fill in what the most accurate description could be. Relativity could and I beleive does play a big role in quantum mechanical behavior, but quantum mechanics is a non-relativistic theory or description of the quantum world. I don't think a relativistic theory of quantum mechanics could ever be realized and that is why there is quantum uncertainty. But, I think it would be far more likely that they could have a similiar description of gravity and that it is just too complex for us to be able to fill in the blancks just yet.
Edit: Removed my comment after fully reading the post.
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LaurieAG
Registered Senior Member
This isn't about whether you think it lame or not but where you anchor your perspective in 'wonderland' or in the real world.
I think it's really ironic that the people who are most likely to be trapped are those who cannot tell the difference between reality and perception.
You're referring to my comment so it is about whether I think it's lame or not lame. It's a lame qualifying comment which is irrelevant to the project to find an equivalence to GR. It's seems to be an attempt to be noticed. That's how the thread started.
I think perception is reality in more ways than we could readily admit. The comment that it is only perception is just a tool to make young physicist happy, but there is no law of physics that says physicist have to be happy. I think the comment was lame and has no place in a scientific paper, as I already explained b does not produce a so then a does not always produce b. So there is no real logic in starting out a paper that way and since it does not produce any logical manner it is unscientific and does not belong there.