Is this a valid way of explaining time dillation?

[His Divine Shadow]

Found this on the net, is this explanation valid?

Think of it this way: everything, everywhere is moving at the speed of light all the time. The thing is, that motion is in 4-dimensional spacetime. As you sit immobile on your ass in front of your computer, your motion is full-speed ahead in the time dimension. Now say you see the ice cream truck going down the street, so you leap to your feet and give chase. As you accelerate, you shift the vector of your motion so that it points slightly into the 3 spatial dimensions and slightly out of the time dimension. The result is that you move faster in space, but slower in time.

As an analogy, think of being in a car headed directly northeast at 100 mph. Part of your velocity is in the east direction and part is in the north; in fact they are both the same (70.71 mph). Now if you turn the car to the right a little, you will still be going 100 mph, but your velocity to the east will be more than the velocity to the north.

[Eframepilot]

Yeah, that does make sense and is mathematically correct, though one normally doesn't speak of motion in the time dimension. The spatial Lorentz contraction factor is exactly the inverse of the time dilation factor, so it really does work. Also, it explains why one can travel both ways in the three spatial dimensions but only one way in the temporal dimension: we're already going unbelievably fast in that "direction".

[drachefly]

While that works mathematically, I think it's easier if you look at it thus (which happens to be both easy and again mathematically equivalent):

How we measure time passing is by the events within our bodies. These events are mediated by electromagnetism, which always moves at the speed of light. Now, if we are moving at a significant fraction of the speed of light, the light has to spend more time catching up than it does going back and forth. This is much like if you're trying to swim across a river -- if it's fast, you have to spend more of your effort swimming upstream.

So, if you're going close to the speed of light, there isn't so much motion left over for your internal changes -- they slow down.

[Kuroneko]

Absolutely. In fact, that is probably one of the best intuitive understandings of special relativity, since it is based on the constancy of the spacetime interval ds²=dt²-dx²-dy²-dz² (the Minkowski metric). Given this interval, special relativity can is equivalent to a single statement: this interval is the same for all observers. It is then quite obvious that to keep the interval constant, a higher spatial distance (dx,dy,dz) must be compensated by a smaller temporal distance (dt), and vice versa--in fact, the way they are related is immediately derivable: since the (spatial) velocity v has v² = [dx²+dy²+dz²]/dt², we have (ds/dt)² = 1-v², i.e., the rate of change of the interval with respect to time (as measured by some stationary observer) is sqrt[1-v²]. Read "interval" as "proper time" and you have the standard time dilation formula.

I disagree with drachefly's interpretation--not in the factual sense (it is, after all, mathematically equivalent), but in that such an intuitive picture paints spacetime structure as a result of electromagnetic behavior rather than the other way around. While historically that was the direction of inference, physically it is backwards.

Also, it explains why one can travel both ways in the three spatial dimensions but only one way in the temporal dimension: we're already going unbelievably fast in that "direction".

No, no, no. In Euclidean space, one can rotate oneself just about any way one wishes, ebut it is impossible to "turn around" in time, true--but the reason for this fact is not any sort of overwhelming momentum "in that 'direction'", but the different sign of the time coordinate in the metric defining the interval. In two-dimensional Euclidean spacetime (t,x), ds²=dt²+dx² (recall the standard distance formula), and the rotation matrix is R = [cos T, sin T; -sin T cos T]. Using the transformation x->ix (T->iT) to make the metric Minkowski (ds²=dt²-dx²) and then cancelling out the imaginary part (cos(iT) = cosh T and sin(iT) = i.sinh T), the Minkowski rotatation matrix becomes L = [cosh T, -sinh T, -sinh T, cosh T]. This is the Lorentz boost--the dilation factor 1/sqrt(1-v²) is cosh T, while v/sqrt(1-v)² is sinh T.

Is it possible to rotate into the past? Well, no. No matter how large the rotation angle T is, a forward-time-pointing vector (e.g, [1;0]) is rotated to a another forward-time-pointing vector (e.g., L[1;0] = [cosh T;-sinh T]). Why is this? In Euclidean spacetime, one can just keep turning around in circles: for a constant ds, ds²=dt²+dx² describes a circle. In Minkowski spacetime, ds²=dt²-dx² describes a hyperbola. The two branches of the hyperbola (or any hyperbola), however, are not connected; it is impossible to travel on one and wind up on the other (unlike the positive-t and negative-t semicircles in Euclidean spacetime).

[drachefly]

It is then quite obvious that to keep the interval constant, a higher spatial distance (dx,dy,dz) must be compensated by a smaller temporal distance (dt), and vice versa

given how the dt^2 term is positive and dx^2 term is negative, don't you need to increase both or decrease both at the same time in order to compensate?

I disagree with drachefly's interpretation--not in the factual sense (it is, after all, mathematically equivalent), but in that such an intuitive picture paints spacetime structure as a result of electromagnetic behavior rather than the other way around. While historically that was the direction of inference, physically it is backwards.

And if people are just trying to get a quick explanation, rather than becoming, say, differential geometry experts or particle physicists, then perhaps it is best to stick to one reference frame at a time, eh? The underlying principles may be more elegant but they are harder to wrap one's mind around.

[Kuroneko]

given how the dt^2 term is positive and dx^2 term is negative, don't you need to increase both or decrease both at the same time in order to compensate?

Mea culpa--I should have reversed proper and coordinate time in what needs to be held constant.

And if people are just trying to get a quick explanation, rather than becoming, say, differential geometry experts or particle physicists, then perhaps it is best to stick to one reference frame at a time, eh? The underlying principles may be more elegant but they are harder to wrap one's mind around.

I don't see why you think so. The explanation at this level doesn't involve anything more than sophomore calculus at worst. Actually, if the differentials are replaced by coordinate differences (which I mentioned via the distance formula reference), it involves only high school mathematics. Moreover, learning to think in spacetime eliminates almost all difficulties with the so-called relativistic paradoxes. The twin paradox of different twin ages, for example, is no more problematic than twin A going from Los Angeles to New York as directly as possible travels a different distance from twin B going from LA to NY but stopping in Canada first. Spacetime is fundamental to relativity; it would be a great disservice not to emphasize it.

[drachefly]

I don't see why you think so. The explanation at this level doesn't involve anything more than sophomore calculus at worst.

And my explanation makes sense to a third grader. I rest my case.

[Kuroneko]

And my explanation makes sense to a third grader. I rest my case.

Yes, by providing third-grader understanding that will largely have to be unlearned. Relativity without a spacetime perspective is effectively neutered, giving rise to difficulties that are completely avoidable (again, the various paradoxes). There are good reasons why physicists consider spacetime fundamental, and this step is not only worthwhile but not nearly as conceptually costly as you make it out to be.

[drachefly]

Odd that you say this perspective gives rise to paradoxes as it is mathematically equivalent to SR. Sure, it doesn't make the transition to GR as graefully, but by the time you're ready for differential geometry I'm sure your perspective is adequately broadened anyway.

And furthermore, this is the perspective suggested by the propagators in quantum field theory, so it's not entirely useless even at a high level.

[Kuroneko]

Odd that you say this perspective gives rise to paradoxes as it is mathematically equivalent to SR.

(1) A spacetime perspective doesn't actually require anything more than the concept of a distance formula and conic sections--high school mathematics. I've stated this before. (2) I have never claimed that there is a mathematical paradox. My only claim was that most cases, such as the so-called twin paradox, are _obviously_ unproblematic from the spacetime perspective--as in, obvious even to a beginner once the concept spacetime is understood, unlike in the space-perspective. This is a significant advantage in understanding relativity's finer points.

[drachefly]

The twin paradox does not even begin to be a paradox so long as you remember to stick to one reference frame... after all, the trouble arises from changing the reference frame without correcting your time coordinate.

[General Trelane (Retired)]

From a lay perspective, the analogy at the start does seem to work, but I see to main problems with it. First, it seems to imply an overly simplistic relationship between time "motion" and space motion. Second, it only works from the PoV of an observer. . .i.e. it depends on the reference frame. Switch to the PoV of the person in motion, and time "motion" from his perspective does not change. Instead, time "motion" of the observer appears to go faster, yet this analogy implies that this cannot happen.

[Surlethe]

From a lay perspective, the analogy at the start does seem to work, but I see to main problems with it. First, it seems to imply an overly simplistic relationship between time "motion" and space motion.

As drachefly pointed out above, the relationship in the analogy is mathematically sound; how is it overly simplistic?

Second, it only works from the PoV of an observer. . .i.e. it depends on the reference frame. Switch to the PoV of the person in motion, and time "motion" from his perspective does not change. Instead, time "motion" of the observer appears to go faster, yet this analogy implies that this cannot happen.

I don't see why the analogy doesn't switch coordinate systems; when you switch from the "observer" to the "mover", the "mover" sees the "observer" moving, which must vector sum with the motion in the time direction to get a magnitude of c.

[General Trelane (Retired)]

My bad. . .I just worked through the math only to find out that it does work. Today is my day to be an idiot, and this applies to #2, which I shouldn't have bothered with at all (it is a matter of reference frames).

[Kuroneko]

The twin paradox does not even begin to be a paradox so long as you remember to stick to one reference frame... after all, the trouble arises from changing the reference frame without correcting your time coordinate.

Indeed--my position is that a spacetime persective not only makes this fact completely obvious but also provides and explicit way of determining how it is corrected. For a layman, the twin paradox looks to be completely symmetrical--after all, the dilation factor is always constant (assuming the turnaround is comparatively negligble, of course). I think the disagreement might be caused by a difference of priorities--yours to answer the question succinctly, and mine to provide a general means to deal with STR questions of such nature. I freely admit I am prejudiced towards geometric means, but for relativity that is completely justified.

[drachefly]

Fair enough. Having never had an occasion to need to describe noneuclidean geometry (including flat minkowski space as noneuclidean) to a non-scientist except for the purpose of relativity, I guess it didn't occur to me that one might have other purposes...