# How can time be a dimension?

In physical terms, dimension refers to the constituent structure of all space and its position in time. Time is same throughout the universe. So, how can time be a dimension?

The point of considering time a dimension is that it transforms as a dimension -- the math in special relativity works out to be exactly that of a 4-dimensional geometry with a Minkowski metric.

The most well-known among laymen example of this is in the invariance of the spacetime interval -- in Galilean relativity, $dx^2+dy^2+dz^2$ is an invariant under rotations as well as boosts -- individually, $dx$, $dy$ and $dz$ may change when you rotate your axes, but their Pythagorean sum remains invariant.

In special relativity, boosts matter, and $dt^2-dx^2-dy^2-dz^2$ is the invariant quantity under this transformation -- this is a generalisation of the Pythagorean theorem to a 4-dimensional Minkowski geometry.

The most general "proof" that time can be considered a dimension similar to those of space, though, is in the full Lorentz transformation, not just of its norm. The Lorentz transformation between $t$ and $x$ (the axis of motion) can be written as:

$$\begin{gathered} t' = t\cosh \xi - x\sinh \xi \hfill \\ x' = x\cosh \xi - t\sinh \xi \hfill \\ \end{gathered}$$

Which takes almost exactly the form of a rotation transformation (it's actually a skew), where boosting into a velocity of $v=\tanh\xi$ is just a rotation in the t-x plane, analogous to $\tan\theta$ for spatial rotations.

• "(I made both dates up, I really don't know the real dates.)" Well, neither date is in the second world war so there is that.. ;) ... and yes it was Hiroshima. – Michael Brown Aug 29 '13 at 15:15
• @MichaelBrown: From a different reference frame! . – Abhimanyu Pallavi Sudhir Aug 29 '13 at 15:16
• You got me there. :P – Michael Brown Aug 29 '13 at 15:17
• For clarity, you might want to add a factor of $c^{2}$ to the time in the invariant interval (so the units are consistent). And in the same vein, you might want to explain why it is that time is not "the same throughout the universe" (one of the main points of modern relativity theory). The questioner probably won't appreciate your answer until they get the relativity of time. – N. Steinle Jul 29 '18 at 13:51
• The $c^2$ is redundant, it is an artefact of using different units for space and time for no good reason (except practicality). – Abhimanyu Pallavi Sudhir Jul 29 '18 at 16:14

Dimension comes from Latin dimensionem "a measuring". We can measure time using a clock and therefore it's a dimension.

• We can measure weight using a spring balance, this is pretty much a really misleading answer. Weight isn't that sort of a dimension. This therefore totally misses the question. – Abhimanyu Pallavi Sudhir Nov 2 '13 at 12:43
• @dimension10 that's right, anything you can measure is a dimension, including weight, mass, pressure etc: dimensional analysis. – Larry Harson Nov 2 '13 at 13:37
• @DIMension10 it looks fine to me, let's leave it at that and disagree. – Larry Harson Nov 2 '13 at 13:39