Time Dilation

Philosophy Forums Science, Technology, and Math
1

I have started on my third edition to the Twin Paradox. And I was going to mention that a proof of the following relationship could be found on Wiki:

The problem is that the Wiki article proves the following:

Notice that the delta t and the delta t’ are reversed.

The first equation shows that delta t is larger than delta t prime. Since delta t prime is the change in time in the moving reference frame, delta t prime should be less than delta t. (Moving clocks should and do run slower than clocks at rest).

To help the reader understand this we should notice the following:

The equation given by the Wiki article on time dilation shows that moving clocks run fast. You can convince yourself of this by setting delta t to 1 and assuming that you can pick a v such that the coefficient of delta t is 100. Then, after 1 year in the rest frame, 100 years could pass in the moving clock frame.

The Wiki article arrives at the equation logically using a small number of geometric relationships. Consider the following:

Mirror C is Mirror A, after moving at a velocity of v for a period of

The Wiki article then draws 3 main conclusions:

After some algebraic manipulations the author arrives at:

Since this equation is wrong, the question is why?

The logic seems sound. The equations 1, 2, and 3 are all simple geometric equations that we should have learned before high school.

But that is the problem!

These geometric equations are all based on Euclidean geometry.

At a minimum we should know that the Pythagorean Theorem requires a Euclidean geometry and in real life this is not the case.

The Euclidean length, L, of an object is given by:

But this equation is not invariant under the Lorentz transform. We really need to use the Minkowski metric where L is given by:

There are actually a number of things that are likely to fail including Eq. 1, Eq. 2 and the apparent parallel paths of the mirrors.

*1 Introduction To Special Relativity by Robert Resnick page 63. It is also implied in Wiki @:

en.wikipedia.org/wiki/Lorentz_factor

*2 en.wikipedia.org/wiki/Time_dilation

2

Where did you get that?
Where did the 2 times come from?

Additionally, the length isn’t contracted in the frame of the inertial observer. Thus the geometry at the base of that triangle, wouldn’t change due to later length contraction issues. The length that contracts is length of the moving object, not the distance it travels.

3

Yeah are you muddling co-moving mass objects equations in with the equation for massless photons. x2 seems kind of odd unless you are talking about two co-moving photons in which case of course it resolves to light speed as the limit obviously and even then it seems an odd resolution to the Lorentz transform to say the least. I don’t see how transformation resolves to times 2 since it is a rotation about an axis or a relation to x and x’ and t and t’.

Chances are if the equation is wrong, then wiki or the author is wrong too, can you explain in more detail? The Lorentz equation as relates to this article is correct as far as I can see, but I may be misunderstanding your position. I certainly don’t get where x2 comes in. Also relativistic mass as discussed in the article is a seldom used term these days because it tends to be misapplied systems where it is not apt. Rest mass is a better term I think and easier to relate to general relativistic concerns as a whole (by which I don’t mean general relativity), not the special concerns of light.

4

Hi Guys,

I am afraid that we might simply have a nomenclature misunderstanding.

The first two equations have asterisks after them in order to provide reference material at the bottom of the article.

If you are referencing something else, could you please highlight it?

Thanks.

P.S. James, I agree with your statements and philosophically believe that there should not be a subject called time dilation. The only change in time that an actual observer would see is a contraction in a moving frame.

Hi Helandhighwater,

I would be happy to try to explain specific comments. Though I fear I might be as bad as Dirac who simply repeated his comments word for word.

Let’s give it a shot!

5

I see what you actually meant was 1. and 2. or (1) and (2) or equation I and II.

I still have no idea what your contention is but there you go.

Whether there should be a subject called time dilation or not, it does agree with experiment. So maybe it is better to say time dilation has experimental evidence: it seems time is dilated at speeds close to light by a significant degree in mass objects particularly (light having its own special case), and length contraction is a consequence therein of the relation of time with space in the transformations, but is more difficult to prove because it is an implicit consequence.

6

Ahh… okay, gotcha

7

Actually, one of the problems that I saw was that velocity is an issue of both time and distance. They granted that the perceived velocity from both reference frames must be the same. If they assert that, the consequence is that if time is to change for a moving object, so must the length, else the velocity would not be perceived as the same.

The question is, why are we presuming that the velocity would be perceived as the same?

There are other issues with the whole relativity thing that it seems either Einstein and crew couldn’t quite work out at that time, or they just chose not to (“Einstein, don’t be telling our god what people are to believe.”).

8

The history of electromagnetism up to that point, perhaps?

9

You could say that about the flat Earth.
…and electromagnetism isn’t the issue, but rather the velocity of any moving object (or more specifically moving observers).

10

Hi James,

At one point I wondered why the velocities should be the same in the primed and unprimed reference frames. There is a simple mathematical reason which I have below:

If you want I can derive the equations:

Thanks Ed

Hopefully everyone will, at minimum, agree that moving clocks appear to run slow.

11

Yes, I am aware of all of that and agree. Thks.

…although the declaration that v = v’ is a questionable choice.

12

Ed3, I’m curious how much you know of vector field analysis…?

13

Hi James,

I am 62 years old, so whatever I tell you should be viewed keeping that perspective in mind.

I graduated from what was then called the Institute of Technology at the University of Minnesota with a Bachelor of Mathematics degree (very unusual). While at the Institute I took two years of Physics, which included a rigorous examination of the Maxwell equations.

I was a mediocre student in physics and a straight A student in math (Except for the D in Partial Differential Equations where I had an attitude problem). This was back in the day when high attrition rates were a badge of honor. The attrition rates for Physics were about 90% per year in the first two years and the attrition rates in mathematics in the Junior and Senior years was about 5 out of 6 per quarter.

I think that I could probably still muddle through, though it is clear to me that my modest case of dyslexia is getting worse.

Thanks Ed

14

Well I seriously heard that. It sounds like we are mutual survivors of the religious war on intellectuals, two pods in the pee, as it were. I suspect we could both tell some serious war stories. And I suspect we have similar neurological injuries, but no vet benefits for it of course.

Although my fields were engineering and psychology with minor in math, I ended up taking all of the courses concerning math offered by UTT and UTA. But in my career, I almost never used any of it. You know where that leads.

In Eugene’s thread, I am introducing Rational Metaphysics and intentionally leaving out as much math as possible for a variety of reasons. RM yields a unified field theory that ends up explaining all of physics mostly through applied definitional logic. But there are places where the math could use some brushing up, just to make the obvious more implacably undeniable. You might want to follow along and mathematically interpret.

Since time dilation is an issue of the motion of objects and tensor field theory involves the very essence of actual cause of motion, I was wondering how comfortable you might be in that arena.

15

Hi James,

I have done a little work with tensors, both as an undergraduate in Differential Geometry,
primarily using the text book “Differential Geometry” by Dirk Struik, and more recently, out of simple curiosity, in Sean Carroll’s book “An Introduction to General Relativity SPACETIME and GEOMETRY”.

However, at least as it applies to Relativity, I do not consider myself “sharp” in this subject matter.

At this point in time I would rather concentrate my efforts into some specific tasks that I feel I need to get done, than work on Rational Metaphysics. Sorry.

Ed

16

S’okay. Such is life.
Everyone is always too busy to thinking about why they are (or aren’t). :sunglasses: