What is the formal definition of spacetime in physics?

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Discussion Overview

The discussion centers on the formal mathematical definition of spacetime in physics, exploring its structure as a manifold and the implications of different metrics in General and Special Relativity. Participants delve into the nature of events within spacetime and the concept of equivalence among them, as well as the significance of coordinate charts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants define spacetime as a pseudo-Riemannian manifold with a specific signature, while others describe it as a pair consisting of a manifold and a locally Lorentzian metric.
  • There is a discussion about whether all events in spacetime are mathematically equivalent or if there exists a privileged '0' element, with some arguing that all events are equivalent in the manifold structure.
  • One participant suggests that in Special Relativity, spacetime is essentially R4, implying that the point (0,0,0,0) holds special significance.
  • Another participant counters that the assignment of (0,0,0,0) as a special point is arbitrary and does not confer any physical significance.
  • Different perspectives on the nature of manifolds are presented, with one participant asserting that not all manifolds are isomorphic to R4, using the example of a sphere to illustrate their point.
  • Participants discuss the implications of choosing a coordinate chart and how this affects the perception of special points within the manifold.
  • References to literature, such as Carroll's lecture notes, are provided to support the discussion on manifolds and coordinate charts, particularly in the context of Special Relativity.

Areas of Agreement / Disagreement

Participants express differing views on the equivalence of events in spacetime and the significance of the '0' element, indicating that multiple competing views remain unresolved. There is no consensus on whether a privileged frame of reference exists.

Contextual Notes

Limitations include varying interpretations of equivalence among events, the dependence on definitions of manifolds, and the unresolved nature of the mathematical implications discussed.

  • #61
@Enrico you have not provided a good reason to reject an axiomatic formulation. In fact, you accepted it for vectors but rejected it for spacetime. Your justification was to capitalize the word THE. That is not a valid justification. Nor is bold or italics or underlining.

Since the professional scientific literature uses axioms then those axioms are acceptable here on PF. Material has been provided using those acceptable axioms as a starting point and reasoning from there. As usual, there are multiple such equivalent starting points and approaches that have been provided.

If you find that unsuitable for your preferences then it is up to you to show that your preferences are also suitable by finding a professional scientific reference that embodies your preferred starting point and construction. If your preferences are incompatible with the literature then your preferences should change.

When you have found such a reference please feel free to open a new thread on the topic. Alternatively, while reading the existing material that has been provided please feel free to open a new thread on any point in any of those references that you find confusing.

As this thread has become unproductive and inconsistent with the professional scientific literature it is now closed. The question in the OP has been answered, and the restrictions that you wish to apply are inappropriate, unjustified, and unclear.
 
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  • #62
Enrico said:
I don't feel comfortable by pointing to the real physical object. If we want to do maths
Adding one additional note: we're not doing math here. We're doing physics. Relativity is physics, not math. To be doing physics and yet not feel comfortable pointing to real physical objects does not make sense.
 
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