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- Thread starter Silviu
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strangerep

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See https://en.wikipedia.org/wiki/Holonomic_basis

The main conceptual difference is that the tetrad vector field constituting a non-holonomic basis is non-integrable, because the respective vector fields in the tetrad do not commute.

Also try googling "object of anholonomicity".

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I don't see anything wrong with your understanding. I'd just add that coordinate basis are useful for calculations, especially of covariant derivatives. Non-coordinate bases are useful for physical understanding. So typically one will do calculations that need them in a coordinate basis, then transform to a non-coordinate orthonormal basis for ease of interpreting the physical significance of the calculations.

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It should also be mentioned that, even if spacetime is, some manifolds are not equipped with a metric. Obviously, orthogonality has no meaning in such a manifold but you can still consider holonomic and nonholonomic bases to your heart’s content.

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I wonder, can the apparent paradox in https://www.physicsforums.com/threads/schwarzschild-from-minkowski.949984/ be resolved in terms of a non-coordinate basis?

See https://en.wikipedia.org/wiki/Holonomic_basis

The main conceptual difference is that the tetrad vector field constituting a non-holonomic basis is non-integrable, because the respective vector fields in the tetrad do not commute.

Also try googling "object of anholonomicity".

Let ##\mu,\nu## be spacetime indices and ##a,b## indices in the tangent space. With tetrads, one usually has

$$\eta^{ab}=e^a_{\mu}e^b_{\nu}g^{\mu\nu}$$

Perhaps the apparent paradox above can be interpreted as a somewhat unusual tetrad formalism in which one has

$$g^{ab}=e^a_{\mu}e^b_{\nu}\eta^{\mu\nu}$$

where ##g^{ab}## are components of the Schwarzschild tensor.

The rational for such an unusual tetrad formalism is this. Given a point in spacetime, one can always choose coordinates in which metric looks like Minkowski near that point. But there is nothing special about Minkowski metric. One can choose coordinates in which metric looks like

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Thank you for this. Ok, that makes sense. I can see why coordinate basis are useful for calculations but could you please elaborate a bit more about why non-coordinate bases are more useful for physical understanding than the coordinate ones (or maybe point me towards some articles)? I am not sure I can easily understand this.I don't see anything wrong with your understanding. I'd just add that coordinate basis are useful for calculations, especially of covariant derivatives. Non-coordinate bases are useful for physical understanding. So typically one will do calculations that need them in a coordinate basis, then transform to a non-coordinate orthonormal basis for ease of interpreting the physical significance of the calculations.

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Well, even if I adamantly insisted that nonholonomic bases are not necessarily orthonormal, you can choose an orthonormal nonholonomic basis. It is then much easier to interpret the resulting vector and tensor components physically.Thank you for this. Ok, that makes sense. I can see why coordinate basis are useful for calculations but could you please elaborate a bit more about why non-coordinate bases are more useful for physical understanding than the coordinate ones (or maybe point me towards some articles)? I am not sure I can easily understand this.

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Suppose, for instance, that you define positions in space by positions relative to a set of satellites that move independently of each other. Each satellite can be thought of as a physical object that defines a local tetrad. This set of satellites is a physical model for a non-coordinate basis.why non-coordinate bases are more useful for physical understanding than the coordinate ones

For a physical model of a coordinate basis you would need a large physical mesh, different parts of which do not move independently. In practice, that would be much more difficult to achieve.

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