# Metric for space-time in SR

## Main Question or Discussion Point

Hi,

I have been wondering if there is a Lorentz-invariant quantity that satisfies the definition of a metric for space-time.

The space-time interval s2 = t2 - r2 [where r is the vector (x,y,z)] does not satisfy the requirement for a metric m that m(t1,r1, t2,r2) = 0 if and only if (t1,r1) = (t2,r2). For instance, ANY two points on the path of a beam of light have a space-time interval of 0. Also, the space-time interval can be either positive or negative, which violates one of the conditions of a mathematical metric.

From what I've been able to dig up, there's some way to map 4-dimensional space-time coordinates to 5-dimensional hyperbolic space and then there is a Lorentz-invariant metric of hyperbolic space, but I couldn't figure out or find what that mapping is (and this might not even be correct in the first place). If anyone could provide a formula for a Lorentz-invariant metric of space-time or point me in that direction, I'd be much obliged!

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bcrowell
Staff Emeritus
Gold Member
Welcome to PF!

I have been wondering if there is a Lorentz-invariant quantity that satisfies the definition of a metric for space-time.
I don't think you want it to be Lorentz-invariant. If it were Lorentz-invariant it would have to be a scalar. The metric is a rank-2 tensor.

The space-time interval s2 = t2 - r2 [where r is the vector (x,y,z)] does not satisfy the requirement for a metric m that m(t1,r1, t2,r2) = 0 if and only if (t1,r1) = (t2,r2). For instance, ANY two points on the path of a beam of light have a space-time interval of 0. Also, the space-time interval can be either positive or negative, which violates one of the conditions of a mathematical metric.

From what I've been able to dig up, there's some way to map 4-dimensional space-time coordinates to 5-dimensional hyperbolic space and then there is a Lorentz-invariant metric of hyperbolic space, but I couldn't figure out or find what that mapping is (and this might not even be correct in the first place). If anyone could provide a formula for a Lorentz-invariant metric of space-time or point me in that direction, I'd be much obliged!
I think the answer is no.

JesseM
does not satisfy the requirement for a metric m that m(t1,r1, t2,r2) = 0 if and only if (t1,r1) = (t2,r2).
Isn't this only a requirement for a Riemannian metric? Spacetime instead is described by a pseudo-Riemannian metric.

Thank you both for your responses, and also for the welcome from bcrowell :)

bcrowell, I might have been a imprecise out of ignorance, but when I say "metric" I really do mean a scalar. From http://en.wikipedia.org/wiki/Metric_tensor" [Broken]:
a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space
So what I mean by "metric" in those terms is the scalar g(v,w), not the tensor that produces it.

Jesse's post was quite to the point, and the comparison of the Riemannian metric and pseudo-Riemannian metric helped to clarify the issue for me. So in light of that, my question might have been more precisely framed as follows: is there any Riemannian metric that can be applied to space-time? (I gather the answer is no.)

My concern about this is mainly physical: if we take the space-time interval to represent something like the "distance" between two events in space-time (an event being defined by its time and position coordinates), and that "distance" can be 0 between apparently different events, doesn't this imply that those events are the same event in some deeper sense? And if we choose not to use the space-time interval as defined above for our "distance" between events in space-time, what should replace it?

I am quite ready to believe that there is no such distance function for space-time (as you've already indicated), but that doesn't make the physical implications any less troubling to me.

Any thoughts would again be much appreciated.

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bcrowell
Staff Emeritus
Gold Member
bcrowell, I might have been a imprecise out of ignorance, but when I say "metric" I really do mean a scalar. From http://en.wikipedia.org/wiki/Metric_tensor" [Broken]:

So what I mean by "metric" in those terms is the scalar g(v,w), not the tensor that produces it.
OK, I just don't think that's what people normally mean by "the metric."

My concern about this is mainly physical: if we take the space-time interval to represent something like the "distance" between two events in space-time (an event being defined by its time and position coordinates), and that "distance" can be 0 between apparently different events, doesn't this imply that those events are the same event in some deeper sense?
No.

And if we choose not to use the space-time interval as defined above for our "distance" between events in space-time, what should replace it?
There's no good candidate to replace it.

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PAllen
2019 Award
Thank you both for your responses, and also for the welcome from bcrowell :)

bcrowell, I might have been a imprecise out of ignorance, but when I say "metric" I really do mean a scalar. From http://en.wikipedia.org/wiki/Metric_tensor" [Broken]:

So what I mean by "metric" in those terms is the scalar g(v,w), not the tensor that produces it.

Jesse's post was quite to the point, and the comparison of the Riemannian metric and pseudo-Riemannian metric helped to clarify the issue for me. So in light of that, my question might have been more precisely framed as follows: is there any Riemannian metric that can be applied to space-time? (I gather the answer is no.)

My concern about this is mainly physical: if we take the space-time interval to represent something like the "distance" between two events in space-time (an event being defined by its time and position coordinates), and that "distance" can be 0 between apparently different events, doesn't this imply that those events are the same event in some deeper sense? And if we choose not to use the space-time interval as defined above for our "distance" between events in space-time, what should replace it?

I am quite ready to believe that there is no such distance function for space-time (as you've already indicated), but that doesn't make the physical implications any less troubling to me.

Any thoughts would again be much appreciated.
For those with a primarily physics background, this definition of metric is a little strange. However, reading through the whole wikipedia entry, I see it really is equivalent to the more traditional formulation. Please read the rest of this article. Toward the end, metric signature and limitations on 'arclength' are discussed quite well.

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From what I've been able to dig up, there's some way to map 4-dimensional space-time coordinates to 5-dimensional hyperbolic space and then there is a Lorentz-invariant metric of hyperbolic space, but I couldn't figure out or find what that mapping is (and this might not even be correct in the first place). If anyone could provide a formula for a Lorentz-invariant metric of space-time or point me in that direction, I'd be much obliged!

http://en.wikipedia.org/wiki/N-sphere

http://iopscience.iop.org/1742-6596/229/1/012038/pdf/1742-6596_229_1_012038.pdf

The usual 4 dimensional Minkowski metric is $ds^2 = x0^2 - x1^2 - x2^2 - x3^2$

where x0 is the time dimension and x1,x2 and x3 are the spatial dimensions. If we propose an additional spatial dimension that is on an equal footing with the other 3 spatial dimensions, then would we get away with simply using:

$ds^2 = x0^2 - x1^2 - x2^2 - x3^2 - x4^2$

?

or if the additional dimension is an additional temporal dimension would this work:

$ds^2 = x0^2 - x1^2 - x2^2 - x3^2 + x4^2$

??

These additional dimensions are not normally detectable, so if they exist, they are probably not on an equal footing with the regular 4 dimensions we are used to and would presumably have to be scaled down in some way.

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atyy