Understanding Minkowski Space Metrics: The Sign Reversal Mystery Explained

[robphy]

I have to agree with Copernicus, I don't see what we gain by pretending that time (or space) isn't imaginary. Depending on how you set up your delta S squared, it could be either one. You can try to explain that away somehow, but I think Minkowski's trick of replacing the Y axis with ict and treating the Lorentz transformation as a rotation is the most instructive approach to understanding the invariant interval.

Minkowski's trick is useful, if you stop with Special Relativity.
If you wish to move to on General Relativity (or do Special Relativity with different coordinates), you'll likely get stuck.
In the end, using the metric $g_{ij}$ is the best way to go.

There is a famous passage in Misner Thorne Wheeler's Gravitation:
"Farewll to ict"
http://www.google.com/search?q="farewell+to+ict"+gravitation

 

[DiracPool]

There is a famous passage in Misner Thorne Wheeler's Gravitation:
"Farewll to ict"
http://www.google.com/search?q="farewell+to+ict"+gravitation

I remember that section, I didn't want to embrace it. But I guess I might have to. GR is really hard to study, I like SR better.

 

[DiracPool]

Btw, I remember reading that section and not being convinced, although now I'm going to have to re-read it. It's frustrating to think that I'm going to have to master GR before I can make a qualified comment on this subject, seeing as it seems obvious that you have to treat either time or space as imaginary. Which one is it?

 

[Fredrik]

it seems obvious that you have to treat either time or space as imaginary. Which one is it?

You definitely don't have to treat any of them as imaginary, since (as I have explained twice) the minus sign(s) can be explained in other ways. And the choice of where to put the minus signs has no relevance whatsoever. You can view the two choices as defining two different theories of physics if you want to, but those theories make exactly the same predictions, so they should at least be viewed as equivalent.

 

[robphy]

This is interesting to note... but it's likely not relevant to the current discussion:

"Where the sign of the metric makes a difference"
Phys. Rev. Lett. 60, 1599 – Published 18 April 1988; Erratum Phys. Rev. Lett. 60, 2704 (1988)
Steven Carlip and Cécile DeWitt-Morette
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.60.1599

(I haven't read it.)

 

[RCopernicus]

You definitely don't have to treat any of them as imaginary, since (as I have explained twice) the minus sign(s) can be explained in other ways. And the choice of where to put the minus signs has no relevance whatsoever. You can view the two choices as defining two different theories of physics if you want to, but those theories make exactly the same predictions, so they should at least be viewed as equivalent.

Your explanation is unsatisfactory. You seem unaware that moving a negative sign into a metric doesn't change the fact that either time or space when squared is negative in the formula for distance in order to create an invariant distance of 0. The only interpretation for this is that either space or time is imaginary.

 

[Fredrik]

Your explanation is unsatisfactory. You seem unaware that moving a negative sign into a metric doesn't change the fact that either time or space when squared is negative in the formula for distance in order to create an invariant distance of 0. The only interpretation for this is that either space or time is imaginary.

Where are you guys getting that idea? It's false. You only have to use imaginary numbers if you insist on using the Euclidean inner product (i.e. the dot product) instead of some other bilinear form. The significance of that minus sign (or equivalently, those three minus signs) is just that a different bilinear form is more useful than the dot product in this theory.

 

[dextercioby]

I don't know if this is worth mentioning, but people outside HEP use the - + + + signature instead of the + - - - one, because not only it allows you to go to an arbitrary number of spatial dimensions without changing the negative value of the determinant, but also because in 4 dimensions the Hamiltonian formulation would mean to disinguish between up spatial indices and down spatial indices. If the signature is + + + on the spatial part, then you can use only <downstair> indices.*

And to make a personal note which seems on topic: I really wish editors would republish some books (valuable ones) whose authors used the ict. The most imporant to me would be: Quantum Field Theory by Umezawa and P.Roman's Theory of Elementary Particles.

*Note: Just after saying that, I noticed that the famous ADM 1962 seem to use + - - - :)

 

[robphy]

Your explanation is unsatisfactory. You seem unaware that moving a negative sign into a metric doesn't change the fact that either time or space when squared is negative in the formula for distance in order to create an invariant distance of 0. The only interpretation for this is that either space or time is imaginary.

As hinted in my post #30
https://www.physicsforums.com/threads/minkowski-space-metric.776078/page-2#post-4884694 ,
with that reasoning, you might conclude something similar [or worse?]
for good-ol' Galilean physics's position-vs-time graph from PHYSICS 101,
which would have $ds^2=dt^2+0^2(dx^2+dy^2+dz^2)$ or ($ds^2=0dt^2+(dx^2+dy^2+dz^2)$, if you prefer the spatial metric).

As Fredrik said, you are led into such mis-interpretations
if you insist on thinking that you are dealing with a Euclidean Geometry [which you are not].

 

[Dale]

The only interpretation for this is that either space or time is imaginary.

This is simply factually wrong. You have been given a different interpretation in posts 2, 3, 4, 29, 36, and 39.

 

[A.T.]

I've never seen a satisfactory explanation of the metrics used in a calculation of distance in Minkowski space. In Euclidean space, the distance is:
ds^2 = dx^2 + dy^2 + dz^2
But in Minkowski space, the distance is
ds^2 = (dt * c)^2 - dx^2 - dy^2 - dz^2
Why are the signs reversed? This implies that space (or time depending on your convention) is imaginary.

If you don't like the negative signs in the metric, you should look into space-propertime diagrams, which are Euclidean:
https://www.physicsforums.com/threads/spacetime-diagram-twin-paradox.671398/page-2#post-4270375

 

[robphy]

If you don't like the negative signs in the metric, you should look into space-propertime diagrams, which are Euclidean:
https://www.physicsforums.com/threads/spacetime-diagram-twin-paradox.671398/page-2#post-4270375

I think it's been mentioned before that such diagrams are not position-vs-time diagrams, as drawn in PHY 101 and in Special Relativity courses.

It's not a faithful mapping of events where each event is represented by one point on the map (as seen on your twins animation... the reunion event appears as two points on the "propertime"-axis).

In addition, light-rays that cross at an event in a Minkowski diagram (i.e. a portion of a light-cone)
are shown as (non-intersecting) parallel segments in the space-propertime diagram.
So, I think a light-clock will look very strange on this diagram.

 

[A.T.]

I think it's been mentioned before that such diagrams are not position-vs-time diagrams, as drawn in PHY 101 and in Special Relativity courses.

Yes, to pros and cons of both type of diagrams are discussed in the thread I linked.

 

[RCopernicus]

This is simply factually wrong. You have been given a different interpretation in posts 2, 3, 4, 29, 36, and 39.

I've yet to read an interpretation in any of these posts that changes the geometry of an invariant distance that's at the foundation of Minkowski space. That is,

0 = (dt * c)^2 + dx^2 + dy^2 + dz^2

The metrics are simply semantic ways to reorganize this relation. At the heart of all of these matrices and formulas is the basic, physical reality that this distance plus this distance plus this distance plus this distance is equal to zero. The only way I know of to make this work is for one of the distances to be negative (or all of the distances to be zero). A negative distance has no real-world equivalent.

 

[Nugatory]

At the heart of all of these matrices and formulas is the basic, physical reality that this distance plus this distance plus this distance plus this distance is equal to zero.

As Fredrik pointed out above, no one is sayng that "this distance plus this distance plus this distance plus this distance" is equal to zero. We are saying that a particular bilinear form on those four quantities (which are not distances, but functions of differences of coordinates) is zero.

It may be time to close this thread, as we are are repeating ourselves without effect.