Metric Compatibility: Is It Forbidden?

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In summary: So it seems that @romsofia and @PeterDonis are saying that a model with a connection that is not metric compatible is mathematically consistent, but not physically plausible. Carroll's notes seem to me to say that we should pick a covariant derivative operator until we find a unique one, which I'm not sure is justified in physical terms.
  • #1
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My question is, is it forbidden to have a connection not compatible with the metric?
 
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  • #2
What do you mean by ”forbidden”?
 
  • #3
Orodruin said:
What do you mean by ”forbidden”?
Senseless, not allowed
 
  • #4
kent davidge said:
Senseless, not allowed
You are not helping yourself by not being more specific. You need to define the context of your question.
 
  • #5
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
 
  • #6
kent davidge said:
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.
 
  • #7
martinbn said:
Yes it is strictly forbidden. It is in the Geneva convention.
Name, rank (both co- and contra-variant parts), and serial number only, right?
 
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  • #8
kent davidge said:
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.
 
  • #9
It is possible to have theories of gravity that arent metric compatible; see Weyl gravity.

The answer to your question is no, connections don't always have metric compatibility.
 
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  • #10
romsofia said:
The answer to your question is no, connections don't 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.
 
  • #11
A clarifying (I hope) question. If I use a different connection on a pseudo-Riemannian manifold, I have a new theory of gravity (that may or may not be physically plausible and/or consistent with experiment) right? That seems to be what @romsofia and @PeterDonis are saying in #9 and #8 respectively.

However, Sean Carroll's lecture notes seem to me to define the covariant derivative by blunt assertion: we keep defining the characteristics we'd like it to have until we've picked out a unique covariant derivative operator (and hence connection - unless I'm missing something). His notes don't really justify these characteristics in physical terms, so I kind of came away with the impression that we were picking it for mathematical convenience. I'm trying to work out if I need to re-evaluate that. Reference: https://preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf, first four pages, in particular on the last one: We do not want to make [zero torsion and metric compatibility] part of the definition of a covariant derivative; they simply single out one of the many possible ones.
 
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  • #13
That turns into quite a reading list if you follow the links you linked, and the links they link. Thanks.
 
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1. What is metric compatibility?

Metric compatibility refers to the ability for different units of measurement to be converted into one another using the metric system. This includes units such as meters, kilograms, and liters, which are based on the decimal system.

2. Why is metric compatibility important?

Metric compatibility is important because it allows for consistency and accuracy in measurements across different countries and industries. It also simplifies conversions and calculations, making it easier to compare and analyze data.

3. Is metric compatibility required by law?

In most countries, the use of the metric system is required by law for trade and commerce. This includes the compatibility of measurements used in labeling and packaging for consumer products.

4. What happens if two units are not metric compatible?

If two units are not metric compatible, it can result in errors and confusion in measurements. This can lead to incorrect data and potentially dangerous situations, especially in fields such as medicine and engineering.

5. Are there any exceptions to metric compatibility?

While the use of the metric system is widely adopted, there are some exceptions to metric compatibility in certain industries and countries. For example, the United States still commonly uses the imperial system for everyday measurements, but even in these cases, there are conversions to metric units available.

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