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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?

Qmechanic
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Tomás
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2 Answers2

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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.

Eric Smith
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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.

ateixeira82
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