Unsimultaneity and quantum physics

[DevilsAvocado]

@DevilsAvocado, regarding Synge.

Yes, of course Synge is right - for one definition of the equivalence principle - the definition which says gravity = acceleration, and that free falling gets rid of the gravitational force. In formal terms, this can be seen from the local cartesian coordinates of a free-falling observer. The spacetime metric in local cartesian coordinates is the same as in flat spacetime for first derivatives, but for second derivatives deviations appear, even at the origin, if spacetime is curved.

Yep, it looks like the second derivative (acceleration) is a 'multifaceted' property...

 

[DevilsAvocado]

Is that your question or a quote from Synge's text?

Good point and bad reference by me, it should have been:

(page 40 in the 1976/1979 edition)

The confusion surrounding the principle of equivalence led J. L. Synge to remark:
. . . I have never been able to understand this principle. . . .

Anyways, the term "gravitational field" is not well-defined mathematically in GR. The Riemann tensor does not codify the "gravitational field".

The more I read about this, the more complex it seems... I'm definitely not in a position to judge, but it looks like there is a Theory of the Gravitation-Field in Einstein's Foundation of the Generalised Theory of Relativity...

It codifies gravitational tidal forces (and tidal torques); it tells us information about shear, vorticity, and expansion of swarms of test particles in free fall (using the exact same mathematical apparatus as fluid mechanics except now in curved space-time). This was actually something Synge first elucidated in his text. The closest thing to the "gravitational field" is the derivative operator $\nabla_{\mu}$ (formally called the Levi-Civita connection); more precisely, the coefficients $\Gamma^{\mu}_{\nu\delta}$ of $\nabla_{\mu}$ are the closest thing to the "gravitational field" and these vanish at the origin of local inertial frames (more precisely: local inertial coordinate systems).

Ahh! Now I see, you mean that the gravitational field (as in electromagnetic field) is not well-defined in its role as mediator of gravitation, as a (direct) force, right? We 'just' get the curved/deformed spacetime, whereas the electromagnetic field is "self-sufficient" in the carrying of electric charges, right? And that gravity in GR is a "fictitious force", or just a 'consequence' of matter/energy following geodetic lines (= no gravitational force), right?

P.S: I like the "fluid part", goes well with my hover!

 

[atyy]

Yep, it looks like the second derivative (acceleration) is a 'multifaceted' property...

Actually, the first derivative is already "local" and "nonlocal". All the derivatives are "local" in the mathematical sense of the limit existing at a point. They are nonlocal in the sense that are first derivative notionally involves the difference between values at two points. In this notional sense, the second derivative is more nonlocal than the first derivative. This is why in physics talk about classical GR, the validity of "free fall = no gravity" provided one ignores "nonlocal measurements" is the same as being able to ignore "second derivatives".

Incidentally, the EP-like idea "acceleration = gravity" is true in the sense that acceleration causes even the first derivatives of the local cartesian coordinates to deviate from flat coordinates. Roughly "acceleration" is "fake gravity" involving the Christoffel symbols (first derivatives of the metric) that WannabeNewton mentioned, and "true gravity" involves the Riemann curvature tensor (second derivatives of the metric).

 

[DevilsAvocado]

(If I'm derailing this thread, please just tell me to stop... )

@WannabeNewton, atyy, bhobba

Maybe it's safest to explain my "curvature outburst" in post #41 ...

First of all – Not Even GNORW!

The naïve 'concept' was this; SR eradicated absolute space/ether and introduced the mathematical abstract flat Minkowski space. GR expanded this 3+1 spacetime with curvature, in the form of the dynamical metric tensor, however still abstract and relative.

Well eh... I just thought the curvature would be something to "hold on to" in finding out "where" you are... in the same way as it's easier to navigate thru Rocky Mountains than the Libyan Desert...

But (of course!) the curved spacetime in GR is still not absolute, and if we imagine two very massive objects (instead of hovercrafts) passing each other in empty space, they will just "drag" their "spacetime bulge" along with them. So when they meet, they will only notice "another bulge" passing by, and no one can say who is moving.

(perfectly clear within the first minute of this video)
https://www.youtube.com/watch?v=MTY1Kje0yLg

What about moving gravitational waves?

Afaik, if we exchange the two objects for two black holes – nothing would change - they will also "drag" their gravitational waves along with them.

What about acceleration?

Well, one obviously will feel the acceleration as a "fictitious force", and others obviously could tell that you are the one moving (in acceleration). End of story.

Almost ...

I found this very fascinating historical paper (that maybe could save my face to some degree): Ether and the Theory of Relativity, Albert Einstein (May 5th, 1920)

My jaw dropped in exponentially for every emphasis (of mine) below.

[13]
More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it, i.e. we must by abstraction take from it the last mechanical characteristic which Lorentz had still left it. We shall see later that this point of view, the conceivability of which I shall at once endeavour to make more intelligible by a somewhat halting comparison, is justified by the results of the general theory of relativity.
[14]
Think of waves on the surface of water. Here we can describe two entirely different things. Either we may observe how the undulatory surface forming the boundary between water and air alters in the course of time; or else — with the help of small floats, for instance — we can observe how the position of the separate particles of water alters in the course of time. If the existence of such floats for tracking the motion of the particles of a fluid were a fundamental impossibility in physics — if, in fact, nothing else whatever were observable than the shape of the space occupied by the water as it varies in time, we should have no ground for the assumption that water consists of movable particles. But all the same we could characterise it as a medium.
[24]
Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.

Phew... it looks like I was not completely way out in my "curvy imaginations"... even though it doesn't work...

Thank you Albert!

 

[DevilsAvocado]

Actually, the first derivative is already "local" and "nonlocal". All the derivatives are "local" in the mathematical sense of the limit existing at a point. They are nonlocal in the sense that are first derivative notionally involves the difference between values at two points. In this notional sense, the second derivative is more nonlocal than the first derivative. This is why in physics talk about classical GR, the validity of "free fall = no gravity" provided one ignores "nonlocal measurements" is the same as being able to ignore "second derivatives".

Incidentally, the EP-like idea "acceleration = gravity" is true in the sense that acceleration causes even the first derivatives of the local cartesian coordinates to deviate from flat coordinates. Roughly "acceleration" is "fake gravity" involving the Christoffel symbols (first derivatives of the metric) that WannabeNewton mentioned, and "true gravity" involves the Riemann curvature tensor (second derivatives of the metric).

Thanks for the explanation atyy!

 

[WannabeNewton]

Even the part where the Christoffel symbols are tensors?

*Runs away*

And that gravity in GR is a "fictitious force", or just a 'consequence' of matter/energy following geodetic lines (= no gravitational force), right?

Right. It's good that you put "fictitious force" in quotes because one must use the term in a liberal fashion when speaking of the "gravitational force" in GR. It isn't the exact same thing as the inertial forces from mechanics (in particular the centrifugal and Coriolis forces). See also exercise 16.5 in MTW.

Take note of the following very important quote from MTW (first two pragraphs of section 16.5):

" 'I know how to measure the electromagnetic field using test charges; what is the analogous procedure for measuring the gravitational field?' This question has, at the same time, many answers and none.

It has no answers because nowhere has a precise definition of the term 'gravitational field' been given-nor will one be given. Many different mathematical entities are associated with gravitation: the metric, the Riemann curvature tensor, the Ricci curvature tensor, the curvature scalar, the covariant derivative, the connection coefficients etc. Each of these plays an important role in gravitation theory, and none is so much more central than the others that it deserves the name 'gravitational field'...Another, equivalent term used for them is the 'geometry of spacetime.' "

P.S: I like the "fluid part", goes well with my hover!

What's a hover ?

 

[atyy]

*Runs away*

Actually, that's one of my favourite parts

 

[DevilsAvocado]

Right. It's good that you put "fictitious force" in quotes because one must use the term in a liberal fashion when speaking of the "gravitational force" in GR. It isn't the exact same thing as the inertial forces from mechanics (in particular the centrifugal and Coriolis forces). See also exercise 16.5 in MTW.

Thanks! Finally I got something right in this "curvature-roller-coaster"!

What's a hover ?

Thank god... you've missed the mess of too big thoughts/pictures in post #41... Don't look!