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I am arguing about this topic with my school teacher in so long time, I want to finish this debate. My teacher's opinion is "Yes, Time is vector" because four-vector has $t$ component, and mine is "No. Time is not vector" here is my counterargument:

  1. proper time $\tau$ is obviously scalar, and since $t=\gamma\tau$ and $\gamma$ is scalar, $\tau$is scalar, so time $t$ is scalar.

  2. if time is a vector, it should have four components, but time is not. Time is just a component of a 4-vector, not a vector itself.

  3. in normal 3-dimensional space, position vector has $x,y,z$ components, but "length" is not a vector, so time is not a vector for the same reason.

And here is teacher's answer,

  1. speed of light $c$ is a constant, so $\dfrac{v}{c}$ is not dimensionless number, so gamma is still vector.

Can anyone answer this problem with clear and strong reasons?

Qmechanic
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kyj519
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  • Like many arguments with teachers, it depends on your definitions. The timelike coordinate is a component of a vector, so it's a vector. But vectors are objects over scalar fields, so it's a scalar. But you could also define the object "$t$" as a timelike unit vector $t$ (For which $t^{\mu}t_{\mu}=-1$), so it's a vector. But in general relativity you can regard coordinates as scalar functions over a patch on your manifold, so it's a scalar. :P It's basically just arguing over a definition. –  Mar 04 '16 at 18:34

1 Answers1

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Time is neither a vector nor a scalar. It is a component of a vector.

A scalar is a quantity that does not change in value under boosts. In Euclidean space, a scalar is a quantity that does not change under rotations, such as distance. Time definitely has different values in different frames, so it can't be a scalar.

But a vector, as you point out, has as many components as the space in which it appears. Four for space time and three for Euclidean space. So time is not a vector.

The proper name is component of a vector.

garyp
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  • Alternatively: yes, time is a vector, in particular it is the tangent vector to your world-line. Or to be more precise: $t$ is the (a) parameter of your world line, and $d/dt$ is the tangent vector (up to factors of $c$). –  Mar 04 '16 at 18:21