Regarding the signature of special relativity

Question

in special relativity we add time as a dimension and replace euclidean space $ \mathbb{R}^4 $ with a pseudo-euclidean space $ \mathbb{R}^{1,3} $ of signature $ (1,3) $ by defining a quadratic form $\eta(x) $ such that

$ \forall x \in \mathbb{R}^{4}, \, \, \, s.t. \,\, x = (ct, \overrightarrow{x}) $

$ \eta(x) = (ct)^2 - (\overrightarrow{x})^2 = \vert x \vert^2 $

which under the assumption that the space is flat becomes the Minkowski metric over which special relativity is constructed.

my question relies on the choice of the signature. why of all the possible signatures, only $ (+,-,-,-) $ and $(-,+,+,+)$ work? what is the physical intuition behind time having a different sign that the other 3? is there a mathematical explanation as to why this works?

Answer 1

The physical intuition behind time having a different sign than the spatial dimensions is our everyday experience that time is not like the other dimensions. We can't, for example, turn around and go backwards in time.

This means that, obviously, spacetime is not $R^4$. But the 3 spatial dimensions do seem to be interchangeable, so they must be treated the same in the metric, leaving time as the odd one out.

Answer 2

You have three spatial dimensions and one time dimension. By analyzing the propagation of a light ray you conclude that you can write (within the usual assumptions of SR)

$$ (ct)^2=x^2+y^2+z^2 $$

Or equivalently

$$ 0=(ct)^2-(x^2+y^2+z^2) $$

If you think in terms of spacetime you will need to define a metric for this mathematical object. Given the previous result for a light ray the natural metric that you would define should have one sign for the time coordinate and the opposite sign for the space coordinates. So you can have $-$ for the time coordinate and $+$ for the space coordinates or the opposite convention.

As @Ghoster already mentioned, usually we group the space coordinates and isolate the time coordinate on the metric but this is just a matter of convenience.

Hence the metric should either have one negative sign and three positive signs or one positive sign and three negative signs. How you order these signs is entirely inconsequential as long as you are consistent with all other definitions and evaluations but it is a lot more convenient to group like coordinates.

Edit: corrected a missing parenthesis.

Answer 3

The choice of signature in the Minkowski metric, whether it's (+,-,-,-) or (-,+,+,+), is deeply rooted in the physical and mathematical principles of special relativity.

1. Physical Intuition: In special relativity, the choice of signature reflects the distinction between space and time. Time behaves differently from spatial dimensions in several aspects. For example, time appears to flow in a single direction (the "arrow of time"), while spatial dimensions do not exhibit such a one-way characteristic. Additionally, the concept of causality is closely tied to the temporal dimension. Therefore, assigning a different sign to time compared to spatial dimensions reflects this intrinsic difference in their behavior.

2. Mathematical Explanation: Mathematically, the choice of signature is related to the structure of the Lorentz transformations, which govern the transformations between different inertial frames in special relativity. The Minkowski metric, with its signature (+,-,-,-) or (-,+,+,+), is invariant under Lorentz transformations. This means that the interval (the square of the spacetime separation) between two events remains unchanged under these transformations.

   Specifically, the Minkowski metric with signature (+,-,-,-) is often preferred because it leads to a "mostly positive" interval for timelike separations (where time dominates), which corresponds to the notion of proper time, while still accommodating spacelike and lightlike intervals. However, the choice of signature ultimately depends on the specific context of the problem and the conventions used.

In summary, the choice of signature in the Minkowski metric reflects both the physical distinction between time and space in special relativity and the mathematical requirements for preserving the invariance of spacetime intervals under Lorentz transformations.

Answer 4

The discussion on the choice of signature in the Minkowski metric is both insightful and essential for understanding the foundations of special relativity. The dual nature of space and time necessitates a careful consideration of their respective behaviors, leading to the adoption of specific sign conventions. Your explanation effectively captures the physical intuition behind this choice, as well as the mathematical framework supporting it. It's crucial for students and enthusiasts alike to grasp these concepts to appreciate the elegance and coherence of Einstein's theory. Well articulated!