Metrics on a manifold, gravity waves, gauge freedom

In summary, the conversation discusses the possibility of introducing a coordinate chart on a finite patch of a manifold, for which a given twice-covariant tensor field can act as a metric. The question is whether this is always possible, and if not, whether it presents a physical answer to the question. The conversation also mentions the relevance of a link provided by atyy, but notes that it does not provide an immediate answer to the question. The necessary conditions for a tensor to act as a metric are mentioned as being symmetric, nondegenerate, and having a Lorentz signature.
  • #1
Michael_1812
21
0
Suppose I have a manifold. I say that it can support a certain configuration of gravity field described by metric tensor \gamma. I do not write \gamma_{\mu\nu}, because that would immediately imply a reference to a particular chart. A tensor field, however, exists on a manifold unrelated to this manifold's parameterisation. It is a box, in which you deposit two vectors, and which then spits out a scalar.

Can I always assume that, for any twice-covariant tensor \gamma, there exists (not necessarily global) a chart, for which this tensor acts as a metric?

Let me now express my question a bit more carefully. We always introduce a manifold via local charts. Suppose some patch of the manifold is parametrisable by some grid z^\mu with metric w_{\mu\nu}. Can I always introduce on this patch a new grid x^\alpha, whose metric is exactly the desired \gamma? (whom I shall now call \gamma_{\alpha\beta} ) Please be mindful that I am talking not about an infinitesimal neighbourhood, but about a finite patch.

I suspect that the answer to my question is negative, though I am not 100% sure. Here is my argument. Suppose that such a coordinate chart can always be assembled. I take a "background metric" \gamma and build the appropriate grid x^\alpha. I take a "perturbed metric" g and construct an appropriate chart y^\alpha. Then, if g and \gamma are "close", their difference being h, I shall say that the difference between g and \gamma is solely due to a coordinate transformation. Hence the gravitational perturbation h is always a gauge effect.

If h can be an actual physical variation, then the answer to my above question must be negative.

This logic, if correct, presents a physical answer to the question.
Still, I would love to see a mathematical argument.

Great many thanks,

Michael
 
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  • #2
http://www.personal.soton.ac.uk/jav/karlhede.html
 
  • #3
Dear atyy,
Thanks for the interesting link. While it is indeed relevant, it still does not provide an immediate answer to the question: given an arbitrary twice-covariant tensor field, is it possible to build, on a *finite* patch, a coordinate grid, for which this tensor field act as a metric?
Thanks again,
Michael
 
  • #4
What do you mean "act as a metric"? Whether a given tensor can be a metric or not doesn't depend on any coordinates. The only condition is that it must be symmetric, nondegenerate, and have a Lorentz signature.
 

FAQ: Metrics on a manifold, gravity waves, gauge freedom

What is a manifold?

A manifold is a mathematical concept that refers to a space that is locally Euclidean, meaning it looks flat when zoomed in, but may have a more complex structure when viewed as a whole. In physics, manifolds are often used to describe the curvature of space-time.

What are metrics on a manifold?

Metrics on a manifold are mathematical objects that define the distance between points on the manifold. They are used to measure the curvature and properties of the manifold, and are essential in the study of gravity waves and other phenomena in physics.

What are gravity waves?

Gravity waves are disturbances in the fabric of space-time caused by the acceleration of massive objects. They are often observed in astrophysics, such as in the form of ripples in the fabric of space caused by the movement of massive objects like black holes.

Why is gauge freedom important in the study of gravity waves?

Gauge freedom refers to the ability to choose different coordinate systems or reference frames to describe the same physical phenomenon. In the study of gravity waves, gauge freedom allows scientists to choose the most convenient and efficient way to describe and analyze the waves without changing their physical properties.

How do metrics on a manifold and gauge freedom relate to each other?

Metrics on a manifold and gauge freedom are closely related, as the choice of metric can affect the gauge freedom of a system. In other words, the chosen metric can determine which coordinate systems are allowed for describing a physical phenomenon, thus affecting the gauge freedom of the system.

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