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I asked this question here. Some doubts emerged in my mind from the comments. Here they are:

Are time and the arrow of time, the same thing? All of modern physics, seems to treat time as just another dimension, except the very important distinction that time has only one direction. So, if there was no arrow of time but time still existed, would we be able to distinguish such a "time" from the spatial dimensions?

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Mathematically, the time dimension would still be distinct from the other dimensions because it is specifically "time-like" rather than "space-like". Our 4-dimensional representation of space-time is that of a Lorentzian manifold rather than a Riemannian manifold. This means the metric tensor that describes distances between points on the manifold has non-positive definite signature. Specifically it has signature of (+---) or (-+++). The difference in signs in this metric is an important distinction between the time dimension and the spatial dimensions.

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  • $\begingroup$ Thank you for your help. So, does this means that space and time are not really at an equal footing in SR? $\endgroup$
    – PhyEnthusiast
    Aug 17, 2018 at 12:22
  • $\begingroup$ Correct, the time dimension is certainly not entirely the same as the other dimensions. However, SR definitely treats time on a much more "equal footing" than a Newtonian viewpoint where time is absolute and space-time has more of a fiber bundle structure. $\endgroup$
    – enumaris
    Aug 17, 2018 at 16:17

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