Does Time Variation Necessarily Imply Full Spacetime Metric?

In summary: MTW discusses the reasons for preferring metric theories in section 38.4. In section 39.2, they state that "All the theories known to be viable in 1973 are metric, except Cartan's." Although they mention that Cartan's theory does have a non-metric theory of gravity, it is not discussed in detail. The observation in the experiment is that there is a time metric, which is experienced directly. If there is time-passage variation, defacto there is a 'non-trivial' time-metric, but a more than one-dimensional context is necessary to give it meaning in terms of 'curvature', so could one indeed say it is a 'non-triv
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wonderingmd
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Background:

Math:
An affine parameter provides a metric along a geodesic but not a metric of the space, for example between geodesics.
A connection provides an affine parameter, and a non-trivial connection gives rise to Riemann curvature.
Given the existence of a connection with Riemann curvature there does not have to be a metric.

Physics: there is time-duration variation on stationary vertically-separated clocks on the surface a mass source (they are not in free fall, so the time is not an affine parameter).

Question: Is this time-effect necessarily part of the geometric structure of space-time, or could it be that it is separate, and there is no full space-time metric- does the existence of this type of effect necessarily imply/require the existence of a space-time metric? So if we measured such a time variation affect alone without even seeing gravitational acceleration or feeling its 'pull' we could know - if we knew enough math - that spacetime is a Riemanian manifold with a metric?

What type of non-purely-gravitational phenomena that we are familiar with in our universe could not exist in a universe with Riemann curvature outside a mass but without a space-time metric?

A string of vertically-separated clocks (separation d) fall freely past a stationary string of vertically-separated clocks (separation d), the times on each are compared, as are the time-lapses between meetings. What is observed?
 
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There are metric and non-metric theories of gravity. Metric theories may also have other fields besides the metric, as in Brans-Dicke gravity and other tensor-scalar theories. MTW discusses the reasons for preferring metric theories in section 38.4. In section 39.2, they state that "All the theories known to be viable in 1973 are metric, except Cartan's."
 
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There is a time metric, we experience it directly, and if there is time-passage variation, defacto there is a 'non-trivial' time-metric, but a more than one-dimensional context is necessary to give it meaning in terms of 'curvature', so could one indeed say it is a 'non-trivial time-metric' even if there is no full space-time metric?

And re the other issues: What type of non-purely-gravitational phenomena that we are familiar with in our universe could not exist in a universe with Riemann curvature outside a mass but without a space-time metric?

A string of vertically-separated clocks (separation d) fall freely past a stationary string of vertically-separated clocks (separation d), the times on each are compared, as are the time-lapses between meetings. What is observed? (it is a way of relating SR time-dilation and GR time-passage-variation)
 

FAQ: Does Time Variation Necessarily Imply Full Spacetime Metric?

1. What is time variation?

Time variation refers to the change or fluctuations in time. It can be observed in various natural phenomena, such as the movement of celestial bodies, the growth of organisms, and the decay of radioactive elements.

2. What is spacetime metric?

The spacetime metric is a mathematical concept used to describe the structure of the universe. It combines the three dimensions of space with the dimension of time into a four-dimensional continuum. It is represented by the metric tensor, which describes the distance between two points in spacetime.

3. Does time variation necessarily imply full spacetime metric?

No, time variation does not necessarily imply a full spacetime metric. Time variation can occur in systems that do not have a complete or continuous spacetime metric, such as in quantum mechanics or in scenarios involving singularities.

4. What is the relationship between time variation and spacetime metric?

Time variation and spacetime metric are closely related, as the metric tensor is used to describe the change in distance between two events in spacetime. In a system with a complete spacetime metric, time variation can be represented by a change in the metric tensor over time.

5. Can time variation occur without a spacetime metric?

Yes, time variation can occur without a spacetime metric. In quantum mechanics, for example, time can be treated as a discrete variable rather than a continuous one. In this case, time variation is described by the change in probability amplitudes rather than a change in the metric tensor.

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