Dirac comment on covariant derivatives

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In summary, Dirac's comments on covariant derivatives highlight their role in extending the concept of differentiation to curved spaces, crucial for maintaining consistency in theories of gravitation and quantum mechanics. He emphasizes the importance of these derivatives in formulating physical laws that remain invariant under coordinate transformations, ensuring that the mathematics aligns with the geometric structure of spacetime.
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Dirac says "Even if one is working with flat space ... one must write one's equations in terms of covariant derivatives if one wants them to hold in all systems of coordinates." But in flat space, covariant derivatives = ordinary derivatives. What does Dirac mean?
Dirac in "General Theory of Relativity" (top of p. 20) says "Even if one is working with flat space ... and one is using curvilinear coordinates, one must write one's equations in terms of covariant derivatives if one wants them to hold in all systems of coordinates."

This comment follows his conversion of the d'Alembert equation $$\Box V = \eta^{\mu\nu}V_{\mu\nu}=0$$ to covariant form $$g^{\mu\nu} V_{;\mu;\nu} = g^{\mu\nu} \left( V_{,\mu\nu}-\Gamma^\alpha_{\mu\nu}V_{,\alpha} \right) =0.$$ In flat space, all the ##\Gamma^\alpha_{\mu\nu}=0##, so covariant derivatives are the same as ordinary (partial) derivatives. For instance, if we work in the Euclidean plane but with polar (curvilinear) coordinates, the metric will change accordingly, but still the ##\Gamma^\alpha_{\mu\nu}=0##.

What exactly is Dirac trying to say here?
 
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The Christoffel symbols are only zero in Cartesian coordinates on flat spacetime. In other coordinate systems (e.g. polars on flat spacetime) they are not zero, so you need the full covariant form of derivatives.
 
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Right! Stupid question. Thank you. (Edit: actually the Christoffel symbols vanish in any system of rectilinear coordinates, not just cartesian...?)
 
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Ibix said:
The Christoffel symbols are only zero in Cartesian coordinates on flat spacetime. In other coordinate systems (e.g. polars on flat spacetime) they are not zero, so you need the full covariant form of derivatives.
And, of course, it's not just the spatial coordinates that you need to consider, it also applies to non-inertial coordinates such as Rindler coordinates (in which an accelerating rocket is at rest) or Born coordinates (in which a rotating disk is at rest). The time-related Christoffel symbols can be interpreted as representing "fictitious forces" such as "g-force" or "centrifugal force" respectively.
 
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Kostik said:
actually the Christoffel symbols vanish in any system of rectilinear coordinates, not just cartesian...?
They definitely don't need to be normalised the same in all dimensions, true, since ##\partial_ag_{bc}=0##. I'm not certain if non-orthogonal lines give zero Christoffel symbols without actually doing the maths.
 
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In rectilinear coordinates the metric entries ##g_{\mu\nu}## are constants (but with off-diagonal terms), all derivatives are zero, so the Christoffel symbols vanish.

After I read your post I considered what ##g## and ##g^{-1}## look like in simple plane polar coordinates, and realized my original post was wrong.
 
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Kostik said:
In rectilinear coordinates the metric entries are constants (but with off-diagonal terms)
Ah yes - then agreed.
 
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Kostik said:
Right! Stupid question. Thank you. (Edit: actually the Christoffel symbols vanish in any system of rectilinear coordinates, not just cartesian...?)
In any affine coordinate system.
 

FAQ: Dirac comment on covariant derivatives

What is a covariant derivative?

A covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. It generalizes the concept of a directional derivative in Euclidean space to curved spaces, ensuring that the derivative of a tensor field is also a tensor field. This is crucial in the context of general relativity and other fields involving curved spacetime.

What was Dirac's comment on covariant derivatives?

Paul Dirac made significant contributions to the understanding of covariant derivatives, particularly in the context of quantum mechanics and general relativity. He noted that covariant derivatives are essential for formulating physical laws in a way that is independent of the choice of coordinates, thus preserving the general covariance of the laws of physics.

How do covariant derivatives differ from ordinary derivatives?

Ordinary derivatives measure the rate of change of a function with respect to a coordinate system, which works well in flat, Euclidean spaces. Covariant derivatives, on the other hand, take into account the curvature of the space or spacetime. They include additional terms involving the connection coefficients (or Christoffel symbols) to ensure that the derivative of a tensor field remains a tensor field.

Why are covariant derivatives important in general relativity?

In general relativity, the laws of physics must be formulated in a way that is independent of the choice of coordinates. Covariant derivatives allow for the differentiation of tensor fields in a manner that respects the curved nature of spacetime. This is crucial for expressing Einstein's field equations and other physical laws in a generally covariant form.

Can you provide an example of a covariant derivative?

Consider a vector field \( V^\mu \) in a curved spacetime with a metric \( g_{\mu\nu} \). The covariant derivative of \( V^\mu \) is given by \( \nabla_\nu V^\mu = \partial_\nu V^\mu + \Gamma^\mu_{\nu\lambda} V^\lambda \), where \( \partial_\nu \) is the partial derivative and \( \Gamma^\mu_{\nu\lambda} \) are the Christoffel symbols. These symbols account for the curvature of the space, ensuring that the derivative is consistent with the geometry of the manifold.

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