General Relativity and the curvature of space: more space or less than flat?

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Discussion Overview

The discussion revolves around the nature of space in the context of general relativity, specifically whether there is more or less space between two points in curved spacetime near massive objects compared to flat spacetime. Participants explore theoretical implications, the concept of curvature, and related phenomena such as time dilation and the Shapiro delay.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether there is more or less space between two points in curved spacetime compared to flat space, suggesting that the intrinsic curvature could imply less space.
  • Others argue that space is a frame-dependent concept and that definitions need to be clarified for meaningful comparisons.
  • A participant mentions that the diameter through a massive object is greater than the circumference divided by pi, implying more space within the same perimeter if mass is present.
  • There is a discussion about the Shapiro delay as an invariant concept that may relate to the question of space and time in curved spacetime.
  • Some participants express uncertainty about whether the geodesic path results in more or less time compared to a "straight" path, with references to specific scenarios involving light travel near massive objects.
  • Concerns are raised about the meaningfulness of comparing distances in curved spacetime to flat spacetime, with examples such as the distance between New York and London on a flat map versus a spherical Earth.
  • One participant seeks confirmation about the constancy of the speed of light as measured by local clocks, even in non-inertial frames, leading to further clarification about coordinate speed versus local measurements.

Areas of Agreement / Disagreement

Participants express differing views on the nature of space in curved versus flat spacetime, with no consensus reached on whether there is more or less space near massive objects. The discussion remains unresolved, with multiple competing perspectives presented.

Contextual Notes

Participants note that the definitions of space and curvature are crucial to the discussion, and the implications of frame-dependence and coordinate systems are acknowledged as limitations in reaching a definitive answer.

  • #61
Martian2020 said:
Thank you. I think I'm understanding better now. Could you please clarify: invariants are e.g. causal relationships, that is clear. Are paths that a hypothetical flash of light would follow through spacetime invariants (I say paths, because light can go many directions, correct?)?
I would say that the path that a flash of light follows is invariant, yes. The coordinates used to describe the set of events on that path will vary from one coordinate system to the next. But the set of events on the path is the same.

However, the angle that a particular light pulse takes from its launching point or the angle at which it arrives at its detection point can vary depending on one's choice of reference frame (e.g. stellar abberation). The path is still an invariant. The angle that it takes when projected onto a spacelike snapshot is not.
 
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  • #62
Martian2020 said:
Thank you. I think I'm understanding better now. Could you please clarify: invariants are e.g. causal relationships, that is clear. Are paths that a hypothetical flash of light would follow through spacetime invariants (I say paths, because light can go many directions, correct?)?
A path through spacetime is not really what we mean by invariant. That's a defined set of points in the spacetime manifold. It's not easy to describe that path until you have chosen a coordinate system, but (and this is the key point), the path exists and is well-defined without being given a coordinate description.

Generally there are two types of path: timelike (followed by massive particles) and null (followed by light). And, there are general timelike and null paths and geodesic timelike and null paths, which are the natural paths that particles and light follow through spacetime. Massive particles can, of course, be forced off geodesic paths, but the path remains timelike. I'm not sure there's any way to force a light onto a null non-geodesic path(?)

There are clearly an infinitude of possible paths, but each particle or light ray can only take one path through spacetime (its worldline).

An invariant is something you calculate, like the length of a spacetime path between two events. Null paths have zero length in all coordinate systems and timelike paths have the same non-zero length in all coordinate systems. So, it's the length of the spacetime path that is invariant.

We don't really talk about the path itself being invariant.
 
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  • #63
PeroK said:
A path through spacetime is not really what we mean by invariant.

It can be. You say:

PeroK said:
the path exists and is well-defined without being given a coordinate description

That's what "invariant" means, so yes, a path through spacetime would be an invariant.

What it would not be is what I would call a "local" invariant, i.e., an invariant defined at a single spacetime point. As you say, it's a set of spacetime points. But that set of points is the same no matter what coordinates you choose.

PeroK said:
I'm not sure there's any way to force a light onto a null non-geodesic path(?)

There is: a waveguide or fiber optic cable are examples of things that can do this.

PeroK said:
it's the length of the spacetime path that is invariant.

We don't really talk about the path itself being invariant.

It's true that the term "invariant" is more likely to be used to describe the arc length along the path than the path itself. However, I don't think that means it's wrong to describe the path itself as invariant; it's just a less common use of the term.
 
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