Hi. I’m under the impression that w/ relativity “space-time” is four space. And four space means that every axis is orthogonal from all the rest. But, if I travel really quickly, the flow of time changes for me relative to others’ frames of reference. But, doesn’t that imply that the two aren’t really orthogonal?

From the traveler’s frame of reference, the flow of time doesn’t change, so they are orthogonal (I guess). But it’s that it is from another frame of reference that the change in time is perceived — in the same way that the length of a moving object shortens relative to an observer in a different frame of reference.

So, that should clear up my confusion, right? To the person moving, traveling doesn’t affect the flow of time, just like moving sideways doesn’t move one forward or back. That’s why we can think of space-time as four dimensions.

More or less. An even better analogy is to consider rotated coordinate systems on a plane. I can set down a set of coordinates and say that “north is this way and east is this way”. You can sit in the same place and set down another set of coordinates that don’t necessarily coincide with mine, even if your north is at right angles to your east. If I then say that a particular tree is due east of us by 100 yards, you won’t agree to that it’s due north, since your “east” is different from mine.

Now replace “north” with “time-direction” and “east” with “space-direction” in the above paragraph, and that’s roughly how special relativity works.

Yeah, I knew that the changing coordinates was involved, in this case it’s not a rotation. I think what screwed me up was confusing the observer w/ the actor. Thanks for the help!

There is an excellent discussion of of this notion in Brian Greene’s The Fabric of the Cosmos. I don’t have it handy so I can’t give you the page number. But I have been reading this stuff as a layman for many years and Greene’s description was the first time I saw this concept described, and what a great job of explaining it.

Suppose I told you i would meet you in the Empire State Building. With the given information, you can narrow the location of our meeting to an x,y coordinate. But you realize now that the Empire State Building consists of 102 floors. You now need a z coordinate to find our meeting place. The room number and the exact floor of the Empire State Building now gives you the precise location. But I never told you what time I was going to meet you there. That is how time is the fourth dimension. A bit less confusing I hope.

For those of you who presume that I stole this, you are right. I took it from a Science Channel special on physics. Can’t remember the episode or anything.

Yes, that is a pretty clear explanation. What may also help to grasp the concept is an understanding that time is used to measure movement relative to other movement. If you compare it to the example above, time doesn’t become a factor if you factor out one of the two people meeting.

Just as you cannot detect movement if you only have one object, you cannot measure movement relative to other movement if you only have one object moving. This is why any time measuring device always adds some form of movement to the movement that is being measured.

Actually, a number of years ago I was researching this very topic, and came across something (lord knows where at this date) that stated explicitly that Einstein’s relativity did away with the notion of time as a fourth dimension.

It puzzled me at the time, and for all I know a serious physicist might regard it as wrong, but I think the reasoning was that to regard time as a fourth dimension, you need to think of it in the same terms as the three space dimensions. That is, two one-inch measures, brought together, will match.

However, due to relativistic time dilation for different bodies under different velocities, the same can not be said. If you compare the measure of one second in both time frames, they will not be equal.

Thus time can not be seen as a fourth dimension in the same way as space dimensions.

This is how I recall the argument going, but it’s been years and my thoughts on it are hazy.

Two different things are going on here: “covariance” and “Lorentz metrics”

First of all, when you bring two one-inch measures together, you’re also implicitly rotating them in space to make them point the same direction, and rotating them in spacetime to make them both travel at the same velocity. Relativity states how things transform as you change your point of view – how they vary together, thus “covariance”. In fact, what you’ve stated is almost exactly backwards: covariance means that we can compare two lengths by putting them in the same frame of reference.

To look at it another way, if the argument about time we correct, then by relativistic spatial transformations the same would go for spatial measurements.

Now, if we ask how intervals (spatial or temporal) are actually measured, we work in analogy from 3-d space: the Pythagorean theorem. If we pick a rectangular coordinate system, then points (x[sub]1[/sub],y[sub]1[/sub],z[sub]1[/sub]) and (x[sub]2[/sub],y[sub]2[/sub],z[sub]2[/sub]) are separated by a length whose square is

When we add time, though, it turns out that the proper formula for the square of the interval between (x[sub]1[/sub],y[sub]1[/sub],z[sub]1[/sub],t[sub]1[/sub]) and (x[sub]2[/sub],y[sub]2[/sub],z[sub]2[/sub],t[sub]2[/sub]) is

The c[sup]2[/sup] is a conversion factor – nothing more, nothing less. Imagine if measurements in the x-direction were in meters and those in the y-direction were in kilometers. Then to get a distance-squared, we’d have to multiply the displacement in the y-direction by 1/1000 (and then square that) to convert to meters. Similarly, t is measured in seconds, so we have to multiply by c (in meters per second) to convert to meters.

That minus sign, though, is a different matter. That means that the metric (way of measuring distances) is “Lorentz”, which is basically saying there’s one sign in it that’s different from the others. That means that temporal displacements are different than spatial ones in that the squares of their intervals are negative rather than positive. Still, as far as the mathematics goes, the two are united in a single 4-d framework.

I cerftainly would never argue with you on a point of mathematics.

My post was an attempt to illustrate the thrust of what I was remembering. Transposition actually played no part in the argument as I recall it. I was merely trying to reconstruct my own thinking on how it made sense.

The main gist was that because of the relativistic effects that bring about phenomena like the “twin paradox”, we must understand time as a purely local phenomenon. As it varies with frame of reference, it can not be used to understand the universe globally in the same way the three spatial dimensions can, and thus does not qualify as a dimension.

I myself actually prefer to think of time as a fourth dimesion, which greatly weakens my ability to argue the other side, but I thought I was duty-bound to at least bring it up.

I like to think, for example, that if space were two dimensions instead of three, and we had a square that grew from a point to a certain size at a constant rate, then abruptly shrunk again at the same rate, it would form an octahedron in 3-dimensional space-time. We could think of ourselves as we change over time as a 3-dimensional cross-section of some 4-dimenstional shape.

The problem with this is that the twin paradox is inherently a GR problem. For the twins to come together again, at least one must accelerate, which removes it from the SR realm. At that point, the methods of measuring spacetime intervals I was describing before become methods of measuring lengths of tangent vectors to a curved spacetime. In that sense, then, all measurements must be done locally, and things may depend on how we move two objects together to be compared. It’s not time-dilation that causes the problem, it’s something differential geometers call “parallel transport” or “the Levi-Civita connection”. Even so, spacetime still is locally composed of four dimensions – three of space and one of time.