B Curve Space-time from Spinning disc?

[lucas_]

In general, unless you already have a good background in the subject of a textbook, you should not try to pick and choose; you should just read the textbook and work the problems.
I would say this is backwards; the fact that the properties of electricity and magnetism (and, as we now know, the properties of all fundamental interactions) are Lorentz invariant, rather than Galilei invariant, is what makes relativity the correct theory, rather than non-relativistic mechanics.
It depends on how you interpret that equation. The best current interpretation is that it is simply an expression of a unit conversion: the conventional units of energy are the conventional units of mass, times $c^2$. In itself this equation says nothing about whether rest mass and other forms of energy can be inter-converted.

It is true that no one has ever developed a non-relativistic theory in which rest mass and other forms of energy can be inter-converted, for example by a particle and antiparticle annihilating each other and producing radiation. However, that could just be due to historical accident--that by the time experiments showed that such reactions were possible, relativity had already been developed and shown to give more accurate predictions in many other experiments, so there was no incentive to try to develop a non-relativistic theory of such reactions. In itself this does not prove that a non-relativistic theory of such things is impossible, or that they would be impossible in a hypothetical alternate universe where relativity was not true.

More generally, claims of the sort Kaku is making are not claims about physics anyway. They are just sensationalistic claims made to sell books, articles, and TV shows. Neither Kaku nor anyone else can do experiments in some alternate universe in which relativity is not true, so nobody knows for sure what would or would not be possible in such a universe.

I heard about E=mc^2 since grader. Do you have any peer reviewed reference(s) exactly how they are derived? Or is it ad-hoc only? About Michio statement "If space and time become distorted, then everything you measure with meters and clocks also becomes distorted, including all forms of mass and energy", it may have a iota of truth in it? I want to see the exact derivations or proof of it. I heard about E=mc^2 since a grader and now need to see it in detail and in full glory. Thank you.

 

[lucas_]

This is a good example of why it's not a good idea to learn science from pop science articles. The article's description is highly misleading. See below.
It's possible that it played a role, but I don't think the connection the article is claiming is valid.

First, a key distinction: the article discusses curved space, but space is not the same as spacetime, and the curvature of the "space" of a spinning disk has nothing whatever to do with spacetime curvature. Spacetime in the case of the spinning disk is flat.

Second, the "space" of the spinning disk isn't even a 3-dimensional spatial "slice" cut out of 4-dimensional spacetime. It's an abstract "space" that you get by performing a particular mathematical operation (the technical term is "quotient space"), and doesn't correspond to any actual 3-dimensional space at all.

Wait. First I know LET discussion is banned (although historical discussion of it not banned). But we are not discussing the details but historical context. I'm just pointing out that Michio Kaku may have it in mind in the analogy of the spinning disc. For if the material physically contract, then in the spinning disc, the inner part contracts more than outside so curvature is actually initiated so here we have LET GR in its full glory? Just state if it is possible. Because Michio Kaku article in Einstein 100 year anniversary is one of the most powerful pop-sci articles about relativity written so it's justified to at least tell the truth whether Michio spinning disc is an example of actual LET GR. If it works, just say yes or no to not violate that details of LET would not be discussed. We are not discussing details of it but I was just asking if Michio example of it can work and historical context.

Remember also when Einstein fist cooking up general relativity. He wasn't using the Minkowski interpretation of space-time. So with regards to Michio statement "One key to Einstein's thinking is to analyze a spinning disk. Since the rim of the disk travels faster than the cetner of the disk, the theory of relativity states that the rim is compressed more than the center. If so, the disk must be distorted" Just see the illustration in the OP. Maybe Einstein really thought of it and it was LET GR in its full glory? Note historical discussion of LET is allowed according to Forum rules. Only details of LET not allowed. So the historical discussion is justified.

Third, the "rubber sheet" model that the article describes is not an analogue of the curved "space" of the spinning disk. Why? Because the curved "rubber sheet" space is an actual 3-dimensional spatial "slice" cut out of the 4-dimensional spacetime of a static gravitating mass. However, this "space" is curved because of the particular way it is cut from the spacetime; there are other ways of cutting such slices that make the spatial slices flat. So once again, the curvature of the "rubber sheet" space is not the same as the curvature of the spacetime; the latter is there no matter how we cut spatial slices out of the spacetime.

 

[Nugatory]

Do you have any peer reviewed reference(s) exactly how they are derived? Or is it ad-hoc only?

Googling for “E=mc^2 derivation” will find some good derivations, and any first-year textbook (such as Kleppner and Kolenkow) will cover this.

Be aware that $E=mc^2$ is a special case (the mass is at rest so the momentum $p$ is zero) of the more general $E^2=(m_0c^2)^2+(pc)^2$. A pretty good rule of thumb is that if whatever you’re reading doesn’t eventually get around to that more general relationship... it’s not giving you the whole story.

 

[phinds]

@lucas_ as others have already pointed out (a point you seem to have taken to heart) your faith in pop-science is very seriously misplaced. I would add to that that your belief in Kaku is even MORE misplaced. He is one of the worst popularizes in terms of reliability and accuracy.

You can find numerous threads here on this site pointing out his flaws. Here are a couple from off-site

http://rationalwiki.org/wiki/Michio_Kaku

http://scienceblogs.com/pharyngula/2011/02/16/why-do-physicists-think-they-a/

 

[Nugatory]

For if the material physically contract, then in the spinning disc, the inner part contracts more than outside so curvature is actually initiated so here we have LET GR in full glory?

No, the spacetime is still flat. The disk is experiencing internal stresses because different parts of it want to move in different directions; either these stresses distort it (as Kaku describes) or it breaks.

 

[Nugatory]

Because Michio Kaku article in Einstein 100 year anniversary is one of the most powerful articles about relativity written...

In the 24 hours that this thread about Kaku has been running, you could have worked your way well into the first chapter of “Spacetime Physics” by Taylor and Wheeler. It’s the difference between looking at a photograph of a delicious meal in a cooking magazine and experiencing the real thing by sitting down and eating it.

 

[PeterDonis]

About Michio statement "If space and time become distorted, then everything you measure with meters and clocks also becomes distorted, including all forms of mass and energy", it may have a iota of truth in it?

Not really. Consider the current SI definitions of the meter and second: the second is defined in terms of a particular hyperfine transition in cesium, and the meter is defined in order to make the speed of light exactly 299,792,458 meters per second. So meters and seconds are defined in terms of a physical process that can be measured anywhere. Since that process defines what meters and seconds are, it doesn't even make sense to ask whether meters and seconds can become distorted.

More generally, suppose you are on Earth and I am in some galaxy a billion light-years away. How could we possibly compare our meters and seconds to see whether they were the same? There is no invariant way of doing that. The only thing we can do is to define our meters and seconds in terms of the same physical process, as SI units do.

 

[PeterDonis]

LET GR

There is no such thing. LET was never extended to cover curved spacetime the way standard Special Relativity was extended to General Relativity.

 

[lucas_]

There is no such thing. LET was never extended to cover curved spacetime the way standard Special Relativity was extended to General Relativity.

Going to your statement: "
First, a key distinction: the article discusses curved space, but space is not the same as spacetime, and the curvature of the "space" of a spinning disk has nothing whatever to do with spacetime curvature. Spacetime in the case of the spinning disk is flat."

But what would happen if the spinning disc is rotating near the speed of light (in convensional SR). Won't the inner rim be more length contracted? What would happen to the spinning wheel when seen at its plane. Won't it create folds in the wheel?

 

[PeterDonis]

what would happen if the spinning disc is rotating near the speed of light.

It wouldn't change anything I said.

 

[lucas_]

It wouldn't change anything I said.

But is not a muon is length contracted and time dilated or any object is length contracted or time dilated when moving near the speed of light. So just want to analyze it in terms of spinning disc at speed of light. Nothing would be length contracted or time dilated? why doesn't it apply to spinning wheel rotating near the speed of light?

 

[PeterDonis]

is not a muon is length contracted and time dilated or any object is length contracted or time dilated when moving near the speed of light. So just want to analyze it in terms of spinning disc at speed of light. Nothing would be length contracted or time dilated? why doesn't it apply to spinning wheel rotating near the speed of light?

I didn't say anything about any of this being wrong. All I said was that spacetime is flat for the spinning disc. That's true regardless of how fast it is spinning. The disc itself being length contracted does not mean spacetime isn't flat; length contraction and time dilation occur in SR, i.e., in flat spacetime.

 

[lucas_]

Adding to the above. Remember in LHC, the particle has time dilation when it spins around the accelerator.. so imagine million of particles (different accelerators) with different radius of particle accelerator superimposed.. each particle would encounter different time dilation and length contraction?

 

[lucas_]

I didn't say anything about any of this being wrong. All I said was that spacetime is flat for the spinning disc. That's true regardless of how fast it is spinning. The disc itself being length contracted does not mean spacetime isn't flat; length contraction and time dilation occur in SR, i.e., in flat spacetime.

Oh. Ok. I'd give it a thought experiment.

Btw. Einstein first thought experiment was racing to go head to head with light. And he couldn't imagine a frozen wave. What principle violated a frozen wave in the first place? Why didn't he just adjust the concept like how Planck had to invent quanta to solve the ultraviolet catastrophe?

 

[PeterDonis]

@lucas_ if you want to go into how the spinning disc is actually analyzed in relativity, I suggest looking up the Ehrenfest paradox and Born coordinates. The Wikipedia articles on these topics give a reasonable starting point:

https://en.wikipedia.org/wiki/Ehrenfest_paradox
https://en.wikipedia.org/wiki/Born_coordinates
There are a lot of complexities lurking here, which require more than a "B" level background. So if you want to discuss this after taking some time to read up, you should start a new thread at the "I" level, not the "B" level. Such discussion is off topic for this thread since we're only talking about the fact that spacetime is flat for this scenario.

 

[PeterDonis]

What principle violated a frozen wave in the first place?

The fact that there is no solution of Maxwell's Equations that describes an electromagnetic wave in free space that varies only in space, not time.

 

[A.T.]

Second, the "space" of the spinning disk isn't even a 3-dimensional spatial "slice" cut out of 4-dimensional spacetime. It's an abstract "space" that you get by performing a particular mathematical operation (the technical term is "quotient space"), and doesn't correspond to any actual 3-dimensional space at all.

The "quotient space" corresponds to the non-Euclidean geometry that one would actually measure by placing rulers on the disc or walking along it with an odometer. So it's not that abstract, but raher quite practical.

 

[PeterDonis]

The "quotient space" corresponds to the non-Euclidean geometry that one would actually measure by placing rulers on the disc or walking along it with an odometer.

More precisely, it's the non-Euclidean geometry you get if you make local distance measurements and then put them together into a global 3-dimensional space. But as I said before, this space does not correspond to any actual 3-dimensional spacelike slice of the 4-dimensional Minkowski spacetime.

The complexities lurking here were discussed in detail in this previous thread from 2014:

https://www.physicsforums.com/threads/the-rotating-disk-of-Albert-einstein.740158/

 

[PeterDonis]

Here is another relevant thread from 2013:

https://www.physicsforums.com/threads/ehrenfests-paradox.693132/

 

[Nugatory]

Adding to the above. Remember in LHC, the particle has time dilation when it spins around the accelerator.. so imagine million of particles (different accelerators) with different radius of particle accelerator superimposed.. each particle would encounter different time dilation and length contraction?

None of the particles encounter any time dilation or length contraction; as far as each particle is concerned, time passes at one second per second.

An observer at rest relative to anyone of these particles (and therefore moving rapidly relative to the surface of the earth) will find that clocks on the surface of the Earth are running slow relative to their own clock, just as an observer on the surface of the Earth will find that a clock at rest relative to the particle is running slow.