Interpreting GR: Is Spacetime Geometry Necessary?

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Discussion Overview

The discussion revolves around the interpretation of General Relativity (GR) as a geometrical theory, specifically questioning whether spacetime geometry is a necessary component for understanding gravity. Participants explore the implications of GR's geometric formulation and its potential limitations, particularly in relation to a future quantum theory of gravity.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants argue that describing GR as "just" a geometrical theory oversimplifies its kinematical aspects, emphasizing that the geometry determines geodesics while also relating curvature to local matter density.
  • There is a distinction made between two notions of curvature: the curvature of a spacetime path versus the curvature of spacetime itself, with implications for how particles move under gravitational influence.
  • One participant cites Einstein's correspondence, suggesting that he did not view GR as purely geometrical, indicating that the distinction between geometrical and other fields is not logically founded.
  • Another participant notes that GR may only be an approximation of a future quantum theory of gravity, which might not involve spacetime geometry at all, raising questions about the applicability of the geometric interpretation at small scales.
  • There is mention of the possibility to redefine the metric in ways that could yield useful theories, although this requires careful consideration of Lorentz invariance.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of spacetime geometry in GR, with no consensus reached on whether GR should be considered purely a geometrical theory. The discussion reflects multiple competing interpretations and ongoing debate.

Contextual Notes

Some limitations are noted regarding the assumptions made about the relationship between geometry and gravity, as well as the potential for future theories to redefine these concepts. The discussion also highlights the unresolved nature of how GR relates to quantum gravity.

kvantti
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Is it justified to state that GR is "just" a geometrical theory to calculate the paths of objects in spacetime?

Just because GR states that "gravity is a property of spacetime itself" doesn't necessary mean that spacetime is curved; just that the paths of objects in spacetime are curved in a gravitational field.

What I'm after is that as we all know, GR is just an approximation of the yet-to-be-discovered quantum theory of gravity, which might not deal with the "geometry of spacetime" at all.
 
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kvantti said:
Is it justified to state that GR is "just" a geometrical theory to calculate the paths of objects in spacetime?

Your statement only describes the kinematical aspect of GR... how the geometry determines the geodesics.
Another aspect [via the field equations] is that [Ricci] curvature is related to the local matter density... figuratively, "matter tells spacetime how to curve". The gravitational field is dynamical.
kvantti said:
Just because GR states that "gravity is a property of spacetime itself" doesn't necessary mean that spacetime is curved; just that the paths of objects in spacetime are curved in a gravitational field.

There are two notions of curvature in this passage.

"A spacetime path is curved" means that the path is not a spacetime geodesic... it has a nonzero 4-acceleration... it is being influenced by something nongravitational. Free particles are inertial and travel on geodesics (with zero worldline curvature).

"A spacetime is curved" means that the Riemann-tensor is not everywhere zero. This means that geodesics may be focusing... figuratively, initially parallel lines may intersect.

Now, it may be that you are trying to suggest that, for example, particle motions are curved in some higher-dimensional noncurved spacetime.
kvantti said:
What I'm after is that as we all know, GR is just an approximation of the yet-to-be-discovered quantum theory of gravity, which might not deal with the "geometry of spacetime" at all.

It may be that the geometric interpretation may be applicable only on classical scales, and may be inappropriate at the small scales.
 
kvantti said:
Is it justified to state that GR is "just" a geometrical theory to calculate the paths of objects in spacetime?

Just because GR states that "gravity is a property of spacetime itself" doesn't necessary mean that spacetime is curved; just that the paths of objects in spacetime are curved in a gravitational field.

What I'm after is that as we all know, GR is just an approximation of the yet-to-be-discovered quantum theory of gravity, which might not deal with the "geometry of spacetime" at all.
This question is rather easy to address since Einstein himself addressed this question. In a letter to Lincoln Barnett on June 9, 1948 Einstein wrote
I do not agree with the idea that the general theory of relativity is geometrizing Physics or the gravitational field. The concepts of physics have always been geometrical concepts and I cannot see why the gik field should be called more geometrical than f.i. the electrodynamic field or the distance between of bodies in Newtonian Mechanics. The notion comes probably from the fact that the mathematical origin of the gik field is the Gauss-Riemann theory of the metrical continuum which we are won't to look at as part of geometry. I am convinced, however, that the distinction between geometrical and other kinds of fields is not logically founded.

Steven Weinberg said something similar to this in his GR text.

Pete
 
kvantti said:
Is it justified to state that GR is "just" a geometrical theory to calculate the paths of objects in spacetime?

Just because GR states that "gravity is a property of spacetime itself" doesn't necessary mean that spacetime is curved; just that the paths of objects in spacetime are curved in a gravitational field.

What I'm after is that as we all know, GR is just an approximation of the yet-to-be-discovered quantum theory of gravity, which might not deal with the "geometry of spacetime" at all.

As I've remarked in other threads, if you have a space-time, AND the usual metric, you can calculate a mathemetical entity (the curvature tensor) which shows that the space-time cannot be flat.

See the https://www.physicsforums.com/showthread.php?t=123922"

thread for more discussion. Note that it IS possible to make such redefintions of the metric and get useful theories - one has to take some extra steps to insure Lorentz invariance when one plays with the metric in this manner, because invariance of the Lorentz interval is no longer automatic if the Lorentz interval is not given by the metric.
 
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