A Einstein and spacetime

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Looking for explanation of Einstein's remarks on geometry/GR
I have seen people using Einstein's comments on the geometrical description of spacetime to mean that he didn't believe in the curvature of spacetime. While I do not think this is true I cannot fully understand what his remarks mean.


When reviewing a book on relativity by Emile Meyerson: La deduction relativiste in 1928 Einstein said:

"I would like to deal more closely with this last point because I have an entirely different opinion on the matter. I cannot, namely, admit that the assertion that the theory of relativity traces physics back to geometry has a clear meaning."

According to the general theory of relativity, the metric tensor determines the behavior of the measuring rods and clocks as well as the motion of free bodies in the absence of electrical effects. The fact that the metric tensor is denoted as "geometrical" is simply connected to the fact that this formal structure first appeared in the area of study denoted as geometry.

"However, this is by no means a justification for denoting as "geometry" every area of study in which this formal structure plays a role, not even if for the sake of illustration one makes use of notions which one knows from geometry. Using a similar reasoning Maxwell and Hertz could have denoted the electromagnetic equations of the vacuum as "geometrical" because the geometrical concept of a vector occurs in these equations."


So he objected to Meyerson's view of GR as geometrical but he does consider curved spacetime as a reality I assume. But I can't explain how he did believe in spacetime but seems to reject the geometrical interpretation? I'm confused?

This paper also explores this issue:

But again I can't get at what exactly it's saying about what Einstein did and did not consider true?
 
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I can't explain how he did believe in spacetime but seems to reject the geometrical interpretation?
I think it's important to conceptually separate the actual physics--the equations, the predictions they make, and the experimental data that confirms the predictions--from the ordinary language words we use to describe the physics, what one might call "interpretation" of the physics (that term is more common in quantum mechanics than GR, but it is a reasonable one to use to describe the difference between "geometric" vs. other viewpoints on GR).

Einstein certainly believed in "curved spacetime" in the sense that he believed that his field equation and its predictions, which include tidal gravity (tidal gravity is the actual physical observation that "curved spacetime" refers to), were correct. He also certainly knew that the mathematics in his field equation could be described as "geometry"; in fact in describing how he came up with General Relativity he described in some detail how he had to learn the mathematics of Riemannian geometry from his friend Marcel Grossman in order to have the mathematical machinery he needed to write down his field equation.

However, it's not as clear how much Einstein accepted the geometric interpretation of GR--by which I mean the view that spacetime is a "real" 4-dimensional geometric object, not just a mathematical model that is used to make correct predictions but is not necessarily "real" itself. It took him a while even to accept that the mathematical spacetime model, which was proposed in 1908 by his former teacher Minkowski, was necessary; it wasn't until about 1912 that he realized he was going to need the math of spacetime and went to Grossman to learn about Riemannian geometry. And the references you give show that even well after GR was finished, he still did not favor the geometric interpretation of the math. The case made by the paper you linked to, that Einstein's favored interpretation was the "field" interpretation, where gravity is viewed as a field much like electromagnetism, is certainly a very reasonable one. (As the paper notes, Weinberg is another well-known proponent of the field interpretation over the geometric one.)

Physically speaking, both interpretations use the same math and make all of the same predictions, so there is no way to distinguish them experimentally. It's a matter of personal preference and which interpretation one finds more acceptable. Einstein was also well known for insisting that physics should be based on actual observable quantities, so I think it's possible that he thought the geometric vs. field interpretation question was not really a question of physics since the actual observable quantities are the same for both.
 
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I see, thank you.
 

pervect

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Summary: Looking for explanation of Einstein's remarks on geometry/GR


According to the general theory of relativity, the metric tensor determines the behavior of the measuring rods and clocks as well as the motion of free bodies in the absence of electrical effects. The fact that the metric tensor is denoted as "geometrical" is simply connected to the fact that this formal structure first appeared in the area of study denoted as geometry.

"However, this is by no means a justification for denoting as "geometry" every area of study in which this formal structure plays a role, not even if for the sake of illustration one makes use of notions which one knows from geometry. Using a similar reasoning Maxwell and Hertz could have denoted the electromagnetic equations of the vacuum as "geometrical" because the geometrical concept of a vector occurs in these equations."
Gravity is different from electromagnetism in that it affects everything, by the equivalence principle. In an electric field, positive charges will be accelerated by the field, neutral charges will not, and negative charges will be accelerated in the opposite direction of opositive charges.

Gravity accelerates everything equally - this is the weak equivalence principle. (There are multiple versions of the principle, but for this argument, the one called the weak equivalence principle is enough). A stone, a feather, wax, lead, and all other substances all fall at the same rate.

Therefore , the geometrical picture is a natural picture for gravity. It's not so natural for electromagnetism - one has to explain how the geometry affects some objects which are charged, and not other objects which are uncharged. THere's no such thing , however, as a gravitationally neutral object , at least not that we've ever discovered. Everything we know is affected by gravity.

The role of geometry in physics really started with special relativity, not general relativity, and was pioneered by Minkowskii. My understanding is that Einstein at first resisted the geometrical interpretation of special relativity, but later on he came to appreciate it's power.

Regardless of the history, the geometrical interpretations of special (and general) relativity is well worth learning. It's worth some serious study. Dismissing the insights that can be gained by the geometrical view on the basis of a lack of information and some casual reading is, to my mind, not well advised. We don't introduce geometry to make General (or special) relativity to make it difficult - we introduce it to make it easier to understand.
 

vanhees71

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I think it's good to have many pictures of the formalism in mind. To my taste nowadays the geometrical aspects are somewhat overstressed in comparison to the point of view by Feynman and also Weinberg that GR is a model of the gravitational interaction as the non-Abelian gauge theories of HEP physics are models for the strong and electroweak interactions. The main difference is that the latter gauge theories use semi-simple gauge groups while GR gauges in a specific sense the Lorentz group as part of the Poincare group. Including spin of particles (e.g., by formulating Dirac spinors in curved spacetime) in this way leads to the extension of GR to Einstein-Cartan theory, i.e., a pseudo-Riemannian manifold with torsion in addition to curvature.

On the other hand the fact that within GR gravitation can be reinterpreted as a dynamical space-time geometry shows the generic difference of gravitational interactions with all others, and the analogy of GR as a gauge theory similar to those underlying the models for all other interactions cannot be taken as far as to a successful quantization. I've no clue, how such a quantized theory of gravitation might look like, but maybe the geometric interpretation is key to understand and finally solve the problem, i.e., to figure out what quantization of the "gravitational field" means in view of the "geometric space-time meaning" of this field.

Of course, for the formulation and use of GR as a classical field theory of gravity it doesn't make all that much of a difference since all of physics is more or less also a quite geometrical discipline, particularly when understanding geometry in a modern extended sense a la Riemann and Klein.
 

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It should be noted that Weinberg later regretted he that had not emphasized more geometric insight in his famous “Gravitation and Cosmology”.
 

pervect

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I think it's good to have many pictures of the formalism in mind.
For the more advanced student, I agree. My concern with those just learning is that too many different approaches and presentations are confusing. This can't always be avoided, though, as there are multiple approaches, especially if we include popularizations that the average PF reader is likely to read. I try to present what I think are the mainstream approaches, which for GR would be the geometric approach.

Occasionally it can be difficult to even identify what the "mainstream" approach is. I believe citation counts could help in theory, but I'm not sure how effective they actually are in settling disagreements about which approach to present first when there are multiple approaches.

In the specific case in question, I do think the "mainstream" approach is geometrical, so that's the one I present. I of course may be biased. And it'd be difficult to back up such a general claim with citation counts, I think.
 

pervect

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An afterthought to what I already wrote. In the case of PF, what I usually try and do as a first priority is figure out what approach the reader has already been exposed to, and give that approach priority.

This doesn't always work as intended. My second priority, after trying to explain what approach I think the poster is pursuing, is to try to explain what I think the "standard" approach is. This is subject to my idea of what a "standard" approach is, but see my previous comments on that point.

Part of presenting a standard approach is to provide a standard reference, if at all possible. For some really basic questions, this isn't necessarily easy to do at an appropriate level.

Occasionally, as a third priority, I think there is some other approrach that isn't standard but would be very helpful. What I try to do in that case (I may not always succeed) is to at least mention the standard approach, mention that I have an alternative that may be a bit non-standard, then present the alternative.
 

vanhees71

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For the more advanced student, I agree. My concern with those just learning is that too many different approaches and presentations are confusing. This can't always be avoided, though, as there are multiple approaches, especially if we include popularizations that the average PF reader is likely to read. I try to present what I think are the mainstream approaches, which for GR would be the geometric approach.

Occasionally it can be difficult to even identify what the "mainstream" approach is. I believe citation counts could help in theory, but I'm not sure how effective they actually are in settling disagreements about which approach to present first when there are multiple approaches.

In the specific case in question, I do think the "mainstream" approach is geometrical, so that's the one I present. I of course may be biased. And it'd be difficult to back up such a general claim with citation counts, I think.
Well, the worst thing for understanding physics is usually to read popular-science books. You have to unlearn almost everything claimed there when starting to learn real physics.

When I had to give an introduction to GR, I'd choose an approach as Landau and Lifshitz vol. 2, i.e., providing the necessary tensor analysis in terms of the Ricci calculus, and of course the first part is geometrical since you have to understand the kinematics first, i.e., how to determine ("infinitesimal") time and space measurements.
 
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However, it's not as clear how much Einstein accepted the geometric interpretation of GR--by which I mean the view that spacetime is a "real" 4-dimensional geometric object, not just a mathematical model that is used to make correct predictions but is not necessarily "real" itself.
A few Einstein quotes might help you:

<< Since there exists in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence. >> (Albert Einstein, "Relativity", 1952).

<< From a "happening" in three-dimensional space, physics becomes, as it were, an "existence" in the four-dimensional "world". >> (Albert Einstein. "Relativity: The Special and the General Theory." 1916. Appendix II Minkowski's Four-Dimensional Space ("World") (supplementary to section 17 - last section of part 1 - Minkowski's Four-Dimensional Space).

<<...for us convinced physicists the distinction between past, present, and future is only an illusion, although a persistent one." >> ( Letter to Michele Besso family, March 21, 1955. Einstein Archives 7-245).

Karl Popper about Einstein:
<< The main topic of our conversation was indeterminism. I tried to persuade him to give up his determinism, which amounted to the view that the world was a four-dimensional Parmenidean block universe in which change was a human illusion, or very nearly so. He agreed that this had been his view, and while discussing it I called him "Parmenides".... >> (Karl Popper, Unended Quest: An Intellectual Autobiography.Routledge Classics. Routledge. pp.148–150).


 
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A few Einstein quotes might help you
One can find Einstein quotes on both sides of the question; note that the OP gives a quote which expresses the opposite view to the ones you give.

If one really wants to investigate this issue, one can't just cherry pick particular quotes. One has to look at all of the available evidence and weigh it. That's what the paper the OP linked to attempts to do. Its conclusion is that Einstein did not favor the geometric interpretation. If you want to try to rebut that claim, you will need to find a reference that conducts a similarly thorough analysis but reaches the opposite conclusion.
 

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