# Metric compatibility

kent davidge
My question is, is it forbidden to have a connection not compatible with the metric?

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What do you mean by ”forbidden”?

kent davidge
What do you mean by ”forbidden”?
Senseless, not allowed

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Senseless, not allowed
You are not helping yourself by not being more specific. You need to define the context of your question.

kent davidge
All texts I read on the topic assume a metric compatible metric. Then I was thinking if this is because a non compatible connection is ill defined or something like that

My question is, is it forbidden to have a connection not compatible with the metric?
Yes it is strictly forbidden. It is in the Geneva convention.

Yes it is strictly forbidden. It is in the Geneva convention.
Name, rank (both co- and contra-variant parts), and serial number only, right?

martinbn
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Senseless, not allowed

As @Orodruin has said, that doesn't help.

If you are asking if a model with a connection that is not metric compatible is mathematically consistent, it is.

If you are asking if a model with a connection that is not metric compatible is physically reasonable, such models don't seem to have worked well so far in matching observations, but AFAIK the question is not completely closed.

If you are asking something other than the above, it should be evident that nobody understands what. So if that's the case, either you need to clarify your question further, or this thread will be closed.

romsofia
It is possible to have theories of gravity that arent metric compatible; see Weyl gravity.

The answer to your question is no, connections dont always have metric compatibility.

kent davidge
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The answer to your question is no, connections dont always have metric compatibility.
To expand on that, it is perfectly possible to have a connection without your manifold having a metric at all so quite clearly metric compatibility cannot be a constraint on a general connection. There are many possible meanings of ”forbidden” or ”senseless” depending on the context of the use of those words. Hence my request for specification.