# About space-time and its four dimensions

I explained to someone I know about General Relativity (as much as I know).

He said that he didn't see how it could be correct.

He argued:

How is 4-dimensional space-time space different to 3-dimensional space? he doesn't agree that as that because the 4-dimensional space-time is only different to 3-dimensions because of the added time dimension. he doesn't think that the added dimension of time would change the space of the system (so 3-dimensional space therefore wouldn't be different to 4-dimensional) because they both still have 3 spacial dimensions, just space-time has an added dimension of time.

I said there will be an explanation to how an added dimension of time would change the spacial structure. is there?

• have you explained Lorentz transformations to him? – luksen May 11 '12 at 23:05
• What is the precise question? This is very broad--- is your friend confused about how time and space mix, or about GR, which has curvature (so that the clocks at different places tick at different rates)? – Ron Maimon May 12 '12 at 3:17
• He's confused as to why space-time has curvature – ODP May 12 '12 at 14:16
• and isn't just like 3-dimensional space – ODP May 12 '12 at 14:16
• Basically, this is the message he sent me: "It talks about adding a dimension because of space-time but i don't really agree with time being associated with any dimensions, let alone time being the fourth dimension. The dimensions are spacial. How odd to have time as one." So suppose he was asking more "how can time be a dimension?" – ODP May 12 '12 at 14:56

Mark's answer is good. Let me just try to put it in what might be simpler terms.

Suppose a flash of light A occurs at a place $(x_1,y_1,z_1)$ and time $t_1$, and another flash B occurs in a different place $(x_2,y_2,z_2)$ and a different time $t_2$.

Suppose $x = (x_2-x_1)$, $y = (y_2-y_1)$, $z = (z_2-z_1)$, and $t = (t_2-t_1)$. So $x$ is the distance between A and B on the x-axis, $y$ on the y-axis, and $z$ on the z-axis.

Now the spatial distance between A and B is $\sqrt{x^2+y^2+z^2}$. That's simple enough - it's just the pythagorean theorem, and it works for any number of spatial dimensions. Now suppose you rotate or twist the x, y, and z coordinate system any way you like. The spatial distance remains the same, even though $x$, $y$, and $z$ are all trading off against each other.

If you want to include time as a dimension just like the others, you'd still like to be able to twist and turn the coordinate system any way you like without changing the 4-dimensional distance between A and B.

It turns out you can't just add $t^2$ under the square root. But you can subtract it. If you use $\sqrt{x^2+y^2+z^2-t^2}$ as the measure of distance, it does work.

Of course, you have to use equal-length rulers on all the axes. If you measure space in feet, you have to measure time in nanoseconds, which is the time it takes light to travel one foot (roughly). Thank Einstein and friends for that.

Yes, adding a true time dimension does affect how the other dimensions "act". The trick is not just adding a dimension of time as if it were some new orthogonal spacial direction.

Instead, the time dimension is added such that it stands in a special relation to the spatial dimensions. Instead of a simple Euclidean 4-space you get the Minkowski space with the symmetry of the Poincaré group and mixed metric signature (−,+,+,+).

This is equivalent to a Euclidean 4-space where one of the coordinates (time) is transformed by multiplying by $\sqrt{-1}$.

This gives a new distance formula (among other things) that has a time dependence. For example, the dimensions of an object depend on the *speed * (distance/time) with which it travels. In other words: the faster something goes, the shorter it gets (Lorentz contraction).

• Mark Beadles has already captured the main point, so I'll just add this. The Pythagorean theorem says that if you have an object that is $x$, $y$, and $z$ units away from you, the "as the crow flies" distance to the object will be given by the famous Pythagorean theorem: $s=sqrt(x^2+y^2+z^2)$. What happened with time was that Einstein and other realized that the Pythagorean theorem almost works the same for time, but with an odd and very important twist: $s=sqrt(x^2+y^2+z^2-t^2)$. See that minus sign on $t^2$? It make time... well... similar but different from space, a "(+++-) signature." – Terry Bollinger May 12 '12 at 2:24