Arclength on a PseudoRiemann Manifold

• Tac-Tics
In summary: Wikipedia gives a confusing definition of a path's length and I would like some clarity. In summary, the length of a path in a pseudo-Riemann manifold is the integral of the distance between the start and end points divided by the speed of light. If the path is timelike, the length is real; if the path is spacelike, the length is imaginary.
Tac-Tics
Wikipedia gives a confusing definition of a path's length and I would like some clarity.

Let M be a pseudo-Riemann manifold with metric g and let a and b be points in M.If y is a smooth function from R->M where y(0) = a and y(1) = b, then it's length is the integral

$$\int_0^1\sqrt{\pm g(y'(t), y'(t))} dt$$

Now, why they decided to put a plus-minus in there, I don't know, but it seems like they MEANT

$$\int_0^1\sqrt{|g(y'(t), y'(t))|} dt$$

Where |.| is the absolute value.

Now on the other hand, this definition of length doesn't distinguish between time-distance and space-distance. It seems like perhaps if we left off the sign-changing nonsense

$$\int_0^1\sqrt{g(y'(t), y'(t))} dt$$

We would get real-valued lengths for space-like paths and imaginary-valued lengths for time-like paths. That's just what my intuition suggests, though. How close would such a guess be to the reality of it?

Why would you want it to be imaginary when you can define it to be real just by changing the sign under the square root? I mean, everything else in the theory is real, so why introduce a complex number unless you have to?

Fredrik said:
Why would you want it to be imaginary

I just want to know the correct interpretation. Is the plus-minus "fixed" at the start of the evaluation of the integral? Or is it chosen at evaluation of every infinitesimal segment (which would simply mean the absolute value)?

The integral is only defined along curves such that the quantity under the square root has the same sign everywhere on the curve. If it's negative, you flip the sign to make sure that you take the square root of a positive number. (So it's the first of the two options you suggested: It's fixed at the start of the evaluation). In relativity this means that the integral is defined for timelike curves and for spacelike curves, but not for any other curves. In the case of timelike curves, we call the result of the integration "proper time" and in the case of spacelike curves, we call it "proper length".

There may be generalizations of this that I'm not aware of. I learned this stuff in the context of general relativity.

Fredrik said:
If it's negative, you flip the sign to make sure that you take the square root of a positive number.

That sounds like you take the absolute value then (which is what I thought). I don't understand why they don't just word it as such on Wikipedia >____>

Thanks =-)

1. What is a PseudoRiemann Manifold?

A PseudoRiemann Manifold is a mathematical concept used in the field of differential geometry that generalizes the concept of a Riemannian Manifold. It is a smooth manifold equipped with a pseudo-Riemannian metric, which is a generalization of a Riemannian metric that allows for the presence of non-positive (or even negative) definite elements in its matrix form.

2. How is the arclength defined on a PseudoRiemann Manifold?

The arclength on a PseudoRiemann Manifold is defined using the pseudo-Riemannian metric. It measures the length of a curve on the manifold, taking into account the non-positive (or negative) definite elements in the metric. This allows for the measurement of curves that would not be possible with a Riemannian metric.

3. What is the significance of arclength on a PseudoRiemann Manifold?

The concept of arclength on a PseudoRiemann Manifold is important in the field of differential geometry, as it allows for the measurement of curves in spaces that may have non-positive (or negative) definite elements in their metric. This is particularly useful in the study of curved spaces, such as those found in general relativity.

4. How is the arclength calculated on a PseudoRiemann Manifold?

The arclength on a PseudoRiemann Manifold is calculated using the integral of the pseudo-Riemannian metric along a given curve. This integral takes into account the non-positive (or negative) definite elements in the metric, resulting in a more general measurement of the curve's length.

5. What are some real-world applications of arclength on a PseudoRiemann Manifold?

The concept of arclength on a PseudoRiemann Manifold has many real-world applications, particularly in the fields of physics and engineering. It is used in the study of curved spacetime in general relativity, as well as in the design and analysis of curved structures, such as bridges and buildings. It also has applications in computer graphics and computer vision, as it allows for the measurement of curves in non-Euclidean spaces.

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