Misleading Phrase: Collapse of wave function

Would this be the thousandth time that idiosyncratic interpretation of 'Einstein's pure GR' has been put here? Yet you have no converts.
For some reason you avoid the temporal aspect of 'inhomogeneous' spacetime, which as various articles explain, accounts for precisely half the total deflection of light passing a massive body. I know you don't accept that - but Einstein did. Have a nice day Farsight - you won't get any f words from me.

 

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Agreement (despite the correction of a typo).

I would like to add that most of what is named "logical fallacy" is reasonable in the logic we need in everyday life, which is not the strong mathematical logic but plausible reasoning.

The logic of plausible reasoning is different. Roughly speaking, if A makes B more plausible, more probable, A is a valid argument in plausible reasoning. But this does not mean that there is a certain rule A -> B, that B holds always if A holds. Usually there is no such logical conclusion. But is it worth to know that A holds, if we want to evaluate if B holds? Of course, all we need here is that P(B|A) > P(A).

Note that plausible reasoning is not diffuse common sense nonsense, but also ruled by precise logical rules - the rules of probability theory. I would recommend Jaynes, Probability theory: the logic of science, about this.

 

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Yep. Let me paraphrase one of Farsight's favourite Einstein quotes......

"......in its physical manifestation, space is not empty, compelling us to describe it by the 10 functions \(g_{\mu \nu}\)"

This of course refers to the metric tensor at every point. Suppose E. was referring to "space-without-time". We can assume then the this tensor acts on a 3-space.

Now the metric tensor is symmetric (as any sensible metric must be), which means that, in its matrix representation it is equal to its own transpose. Then, since the entries on the principal diagonal are invariant under transposition, and by symmetry the 6 off-diagonal entries are equal in pairs, this leaves 3 + 3 = 6 independent entries.

If on the other hand we assume it acts on a 4-space, then there will be 4 + 6 = 10 independent entries, as Einstein claimed. We may assume this 4-space is spacetime

 

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Didn't I see you claiming earlier that there is no spacetime curvature because Einstein didn't like it?

You can call it "inhomogeneity," you can call it "curvature," like I said before it's a distinction without a difference.

Try to be consistent, please.

 

I don't see why there should be any confusion about what "curvature" is in GR. At the end of the day, "curvature" is a piece of terminology and it's not too hard to find out what physicists in the GR community mean by it. If you do that you'll find out that there are a small handful of so-called curvature measures. The most general of these is a quantity called the Riemann tensor, and when GR researchers talk about spacetime being "curved", they generally more precisely mean "the Riemann tensor is nonzero".

Where the Riemann tensor is nonzero, I'd have thought that the important thing isn't what this situation is called but what it means, i.e., what it implies in the context of GR. A couple of well known implications of the Riemann tensor being nonzero are:

1. It is impossible to map all of spacetime with a homogeneous coordinate system. (More mathematically: the metric components \(g_{\mu\nu}\) are necessarily inhomogeneous, i.e., they cannot be constants over all of spacetime, independently of the choice of coordinates.)
2. Geodesic trajectories near each other that are initially parallel will generally not remain parallel but will tend to converge or diverge (in a way expressed mathematically by the Jacobi equation).

Einstein described the Riemann tensor and pointed out both properties of it above in, for instance, his 1916 paper on GR. The definition he uses and what he says about it there are, in every important respect, identical to what you would find in a more recent textbook or what you would be told in a decent university GR course. As for everyone calling this "curvature", most physicists and mathematicians just intuitively think of points 1 and 2 (among others) as telltale signs of a curved space (in general), so it gets called "curvature".

These points are easy enough to understand if you think of a familiar curved surface like the surface of the Earth:

1. It's well known that you can't draw a (flat) map that faithfully represents sizes and shapes all over the Earth. For instance, this map makes countries very far north and south look much wider and larger in area than they really are compared with countries near the Equator, and it depicts the North and South poles stretched out to entire lines. This intuitively illustrates what is meant by a (coordinate) mapping being "inhomogeneous" in point 1 above.
2. Lines of longitude (meridians) are examples of geodesics on the Earth's surface (ignoring for simplicity that Earth isn't a perfect sphere). Nearby lines of longitude are parallel at the Equator and converge at the poles.

 

Farsight: My question from posts 109/115 is now a couple of pages back, so I'll reiterate it.

According to your view, what does the physical field look like for two widely-separated photons with maximally entangled polarization?

 

Whether light goes straight or pencils fall down depends not on whether space is homogeneous but depends on the observers frame of reference. E.g. consider an inertial frame in flat spacetime. In such a spacetime space is flat. In this frame photons travel in a straight lines and objects subject to no other force than the gravitational force will remain at rest or move in a straight line. If the observer now changes to an accelerating frame photons will not follow curved paths and free objects will "fall down".

 

For the most part you're correct. However I've seen people being sloppy on this. One text spoke of the universe being flat. When this is said it means that space is flat but spacetime is curved. It's a bit tricky in this instance. There's a difference between flat space and flat spacetime. When spacetime is flat then this is what GRist will mean when the Riemann tensor is zero.

Nicely stated. It's refreshing to see people posting nowadays who know what they're talking about when it comes to GR.

There is one point in GR that Einstein viewed different than the majority of modern GR experts. If you read Einstein's GR paper of 1916 that you mentioned above very carefully then you should have noticed that there's nothing in it which states that gravity is a curvature in spacetime. To Einstein the presence of a gravitational field was determined by the inertial acceleration of a free object. By this I mean if you drop a pencil and if falls then you're in a gravitational field. And this is regardless of whether there's any spacetime curvature or not. As you know spacetime curvature and tidal forces are two ways of expressing the exact same thing. That's how Einstein was able to state that a uniform gravitational field is equivalent to a uniformly accelerating frame of reference. Einstein himself stated in no uncertain terms the existence of a gravitational field is determined by the non-vanishing of the components of the affine connection, not the vanishing of the no vanishing of the[components of the Riemann tensor.

 

Schneibster - Having been absent from here for a long time there's something I'd like to learn about this forum. Would you mind helping me in PM (or what passes for PM here)? Thanks.

 

Sorry, PMB, I'm pretty new and am not likely to know what you want, and also I don't have PM privileges yet.

 

I own a physics forum. Membership is by invitation only. If you ever become interested in joining please let me know.

 

You describe the state of space using the ten functions. The Earth is surrounded by space, not spacetime. There's no motion in spacetime. And as I said previously, inhomogeneous space is the reality that underlies what you think of as curved spacetime. Which is why Einstein said this:

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. 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."

No, I referred to http://arxiv.org/abs/physics/0204044 :

There exists some confusion, as evidenced in the literature, regarding the nature of the gravitational field in Einstein's General Theory of Relativity. It is argued here the this confusion is a result of a change in interpretation of the gravitational field. Einstein identified the existence of gravity with the inertial motion of accelerating bodies (i.e. bodies in free-fall) whereas contemporary physicists identify the existence of gravity with space-time curvature (i.e. tidal forces). The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with.

Unfortunately there is. I think it goes back to Wheeler.

No problem there.

However some things are not identical. For example you won't find a recent textbook saying a gravitational field is a place where space is neither homogeneous nor isotropic. Or a place where the speed of light is spatially variable.

The trouble with that is that a gravitational field is a place where space is inhomogeneous, not curved.

This can be misleading in that it makes people think that space is curved in some higher dimension. It isn't. We perhaps need a new thread on this.

 

I'm afraid I have to disagree with that Pete. Einstein made it clear that space is real, it isn't nothing, and he described a gravitational field as a place where space is neither homogeneous nor isotropic. Your frame of reference is an abstract thing. It's little more than your state of motion. Your motion through space doesn't change space. When you're accelerating, your pencil might fall down and you could say there's a gravitational field of sorts. But it isn't the "special form" of gravitational field that can't be transformed away. That's the sort of gravitational field I'm referring to. The Earth's gravitational field. See section 20 of Relativity: the Special and General Theory where Einstein said this:

“We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes”.

 

I give up - no sort of reasoned argument will alter this guy's determination to be wrong, and life is too short to continue trying.

 

it will be one of the biggest mistakes you will have done to yourself.. it is wise to stay away from peter. i suggest NOT pm-ing him.

 

It's not peer reviewed.

It's an opinion. I've given some pretty good reasons to believe that it's wrong. I'll repeat one of them:

\(R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}\)

Every term on the left side of that is a curvature, and was called that since before Einstein used it.

The equations (ten of them) that follow from that are called the "Einstein Field Equations" and are the field equations of the gravity field.

Next?

 

Would you mind contact me at e-mail address? It will be very clear why I want to speak with you if you do. My address is a Gmail account using my current username. Thanks.

 

I did notice the author of the paper F keeps posting. Just sayin'.

 

I'm sorry, I don't think I'm interested. Thanks for the offer.

 

Einstein never described a gravitational field in that way. Where do you think/assert that he said that?

Of course. That's when the spacetime is curved, i.e. there are tidal forces present. However that's not how Einstein defined the gravitational field. See
http://wikilivres.ca/wiki/The_Foundation_of_the_General_Theory_of_Relativity

And that is precisely what Einstein said, to the letter.

All that means is that when Einstein was talking about transforming the gravitational field away that it can't be done in general for the entire spacetime if the spacetime is curved. However for all spacetimes, whether curved or flat, the gravitational field can always be transformed away in a finite region - And that holds in all possible circumstances.