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In describing the length of a curve in spacetime it is necessary to parametrize it and in GR one comes across the notion of affine parameters.

One definition of affine parameters u are parameters which are related 'affinely' to the length of the curve s trough u = as + b where a and b are constants. These parameters has the property that the tangent vector to the curve remain constant in length - however spacetime is a psaudo-Riemannian manifold, so for null curves which have zero length (and are crucial in the description of photons) one can not use affine parameters to parametrize these curves since the length does not vary along them.

Generally a geodisic can be defined as a curve for which the tangent vector remain parallel to itself i.e.

$$\frac{d \vec t}{du} = \lambda(u) \vec t$$

and another definition of affine parameters is to parametrize the curve such that

$$\frac{d \vec t}{du} = 0.$$

But this implies that also the length of the tangent vector remain constant, so I worry that this definition of an affine parameter is equivalent with the old one which could not be used for null curves. Is it possible to find parameters which are not related by to the length of the curve, but which still make the length of the tangent vector constant along the curve? If so does anyone have an example of such a parameter?

Or to state my question more clearly: Does one have to use an affine parameter in order to satisfy the geodisic equation

$$\frac{d \vec t}{du} = 0.$$

If not is this somehow obvious? Any examples?

One definition of affine parameters u are parameters which are related 'affinely' to the length of the curve s trough u = as + b where a and b are constants. These parameters has the property that the tangent vector to the curve remain constant in length - however spacetime is a psaudo-Riemannian manifold, so for null curves which have zero length (and are crucial in the description of photons) one can not use affine parameters to parametrize these curves since the length does not vary along them.

Generally a geodisic can be defined as a curve for which the tangent vector remain parallel to itself i.e.

$$\frac{d \vec t}{du} = \lambda(u) \vec t$$

and another definition of affine parameters is to parametrize the curve such that

$$\frac{d \vec t}{du} = 0.$$

But this implies that also the length of the tangent vector remain constant, so I worry that this definition of an affine parameter is equivalent with the old one which could not be used for null curves. Is it possible to find parameters which are not related by to the length of the curve, but which still make the length of the tangent vector constant along the curve? If so does anyone have an example of such a parameter?

Or to state my question more clearly: Does one have to use an affine parameter in order to satisfy the geodisic equation

$$\frac{d \vec t}{du} = 0.$$

If not is this somehow obvious? Any examples?

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