Minkowski Metric and the Sign of the Fourth Dimension

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Discussion Overview

The discussion centers on the properties of the Minkowski metric, particularly the sign of the fourth dimension unit vector representing time in Minkowski space. Participants explore the implications of different signatures for the metric, including the standard (-,+,+,+) and (+,-,-,-) signatures, and their relation to the geometry of spacetime and the behavior of light in various reference frames.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question why the time unit vector in Minkowski space is opposite in sign to the spatial unit vectors, suggesting it may fulfill certain geometric requirements.
  • Others propose that a mixed signature space has unique geometric properties, such as invariant directions under Lorentz transformations, which align with the behavior of light rays.
  • A participant asserts that the negative time unit vector is necessary for the ten-dimensional Poincare group to include rotations and boosts, while another challenges this by stating that the isometries remain ten-dimensional even with a (+,+,+,+) signature.
  • Some participants discuss the experimental basis for the Minkowski signature, emphasizing that it ensures the constancy of the speed of light across reference frames.
  • There is speculation about whether a mathematical demonstration exists to show that a (+,+,+,+) signature would not maintain the speed of light as constant in all frames, contrasting it with the Minkowski signature.
  • Participants explore the concept of invariance of vectors under rotations, noting that Minkowski space allows for lightlike paths that are invariant, unlike Euclidean space.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the Minkowski signature and its relationship to the properties of spacetime and light. There is no consensus on the necessity of the negative time unit vector or the dimensionality of the isometry groups associated with different signatures.

Contextual Notes

Discussions involve assumptions about the nature of spacetime and the definitions of vectors, as well as the implications of different metric signatures on physical laws. Some mathematical steps and definitions remain unresolved.

Epistimonas
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Why is the unit vector for time in Minkowski space i.e. the fourth dimension unit vector always opposite in sign to the three other unit vectors?

The standard signature for Minkowski spacetime is either (-,+,+,+) or (+,-,-,-).

Is there some particular reason or advantage for making time opposite to the spatial dimensions?
 
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A mixed signature space has certain geometric properties different from ordinary, Euclidean space. One of them is the presence of directions that don't change regardless of "rotations" that mix up space and time. These "rotations" are Lorentz transformations, and these directions that are unchanged are the paths of light rays. Minkowski space is the only flat space with this property* being in line with what we know about relativity.

*You can have a (0,+,+,+) space where the timelike vector is invariant, but you can show this corresponds to Galilean invariance, which we know not to be present in the physical world.
 
Right okay that is a grand answer to my question, thank you. I am familiar with the Poincare group that contains the isometries of Minkowski spacetime. Essentially the answer you gave me was, "The negative time unit vector is required to fulfill the requirements of the 10-dimensional Poincare group, consisting of a translation through time, a transition through any 3 directions of space, a rotation around any of the three spatial axes, and a boost in any of the three directions."

If the time unit vector was not negative the the group of isometries of the spacetime of metric (+,+,+,+) would only be 7-dimensional, lacking in the rotation around any of the spatial axes.

Is this correct?
 
No, the group of isometries in (++++) signature is still ten-dimensional (6 rotations in the 6 orthogonal planes, and 4 translations).

The reason spacetime has the Minkowski signature (-+++) is because of the experimental fact that light has the same speed in every reference frame. Lorentz transformations can rotate among the 3 spatial directions, or they can mix time and space (via 3 boosts), but they always leave the lightcone intact.
 
If the time unit vector was not negative the the group of isometries of the spacetime of metric (+,+,+,+) would only be 7-dimensional, lacking in the rotation around any of the spatial axes.

Hm, I don't think so. Even a 4d Euclidean space would have 4 translational and 6 rotational degrees of freedom. The point I was trying to make was that the Minkowski signature is chosen because, in general terms, out of the three possibilities for a flat space--Euclidean, Galilean, and Minkowskian--only the latter has the correct notion for the principle of relativity. The others can be ruled out experimentally.

If you're asking from a pure math standpoint, then I'm not sure I understand the question.
 
Ben Niehoff said:
The reason spacetime has the Minkowski signature (-+++) is because of the experimental fact that light has the same speed in every reference frame. Lorentz transformations can rotate among the 3 spatial directions, or they can mix time and space (via 3 boosts), but they always leave the lightcone intact.

Okay so there must be a mathematical way of showing that for a (+,+,+,+) signature the speed of light is not the same in every reference frame, and that for a (-,+,+,+) or it's equivalent the speed of light has a constant speed in every frame?

And if that can be shown, then it could then be shown that the constant that the speed of light travels at is equivalent to the speed a massless particle travels?
 
Epistimonas said:
Okay so there must be a mathematical way of showing that for a (+,+,+,+) signature the speed of light is not the same in every reference frame, and that for a (-,+,+,+) or it's equivalent the speed of light has a constant speed in every frame?

And if that can be shown, then it could then be shown that the constant that the speed of light travels at is equivalent to the speed a massless particle travels?

The key is to show that, under generalized rotations, a Euclidean space has no invariant vectors in the plane of the rotation, whereas a Minkowski space does.
 
Okay so is what you're saying that the definition of a vector is that it is invariant under translation and rotation, and that in Euclidean space, vectors are not invariant under rotation and that in Minkowski space they are?
 
Definition of a vector, no. Obviously not every vector is invariant under rotations. But for there to be lightlike paths, there must be some vectors that are invariant under rotations, even when that vector is in the plane of rotation.
 

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