Time Dilation & Length Contraction, Further Thoughts

[Nugatory]

One last question about rest frames, squrt 1 - v2/c2 tells us that at c T becomes infinite and length 0. please explain how something taking infinite time to move 0 distance is not at rest.

The time dilation and length contraction equations actually tell us no such thing, because they do not apply when v=c. They are derived using a sequence of mathematical steps that starts with the (usually implicit) statement "If v<c, then we can show that..." and proceeds from there.

Consider the formula for the area of a circle: A=\pi{r}^2. It's pretty clear that if I shrink the radius towards zero, the area of the circle shrinks to zero as well. But suppose I were to keep shrinking the radius to even smaller (negative) numbers... I could say that the formula "tells us" that the area of the circle starts to increase as the radius shrinks beyond a certain point. But just about everyone reading this (including you, I hope) would rightly tell me that the radius of a circle is always non-negative so I just plain can't attach any meaning to the area of a circle of negative radius.

 

[Dale]

One last question about rest frames, squrt 1 - v2/c2 tells us that at c T becomes infinite and length 0. please explain how something taking infinite time to move 0 distance is not at rest.

Again, there is no reference frame moving at c. See post 36.

You keep trying to start your reasoning from a false premise, and then seem surprised that you wind up with problems.

 

[Adrian07]

Thanks for the clearer explanations they basically mirror what I have been thinking, I think the confusion lies in the fact that we have been approaching from different directions. I see time as vertical and distance as horizontal but any forward movement as left to right. I did realize that light seconds were a unit of distance. Thanks for all the communication even if I do not appear to have understood at times it has helped me think.
I now believe that I have worked out why the formulas take the form they do, in fact I now know where the answers to the 3 paradoxes of relativity lay, that is the pole and barn, the twins and the grandfather paradox. The answers to these show where relativity is not completely understood, there are no paradoxes.
For the record T = infinity and distance (length) = 0 do make sense but not within the structure of the universe they actually tell of infinite energy in perfect symmetry with a perfect vacuum, in other words what existed before the BB, although I have no doubt you will disagree.

 

[ttakacs]

Hello to everyone,

I have a question addressing both special relativity and quantum mechanics. On one hand from special relativity we know that the longitudinal length of a moving object decreases with increasing speed, as much as goes to zero with getting close to c.

However on the other hand, at the opposite order of the scale we have Planck length - a very very small value but definitely greater than 0 meter. If this is an absolute lower limit of length, than introducing it to the relativity formula we get a limit speed slightly less than the speed of light, but a new theoretical limit for velocity.

Maybe the main issue whether Planck length is a real lowest limit or just a limit for our observation and there are smaller distances nevertheless cannot be experienced.

Please give me an explanation or resolve this apparent paradox. Thank you.

 

[PeterDonis]

If this is an absolute lower limit of length, than introducing it to the relativity formula we get a limit speed slightly less than the speed of light, but a new theoretical limit for velocity.

The Planck length is a quantum phenomenon, plus it involves gravity; so it isn't covered by special relativity for two reasons, so to speak.

As far as how the transformation laws between frames would have to change if the Planck length were an absolute lower limit on length, just changing the "speed limit" to slightly less than that of light would not be enough by itself, because that "limit" would not be frame-invariant. There is a theoretical proposal called Doubly Special Relativity which modifies the Lorentz transformations at short distances so that the Planck length is frame-invariant (i.e., an object that looks 1 Planck length long in one frame looks 1 Planck length long in all frames).

Another alternative would be to simply say that ordinary relativity stops being valid at short enough distance scales, and some other physics (such as string theory) takes over. This would have to apply to both special and general relativity, since, a I noted above, the Planck length involves gravity.

We can't do experiments at anything like the distance scales we would need to to start testing these alternatives; the Planck length is something like 20 orders of magnitude smaller than the smallest scales we can probe experimentally at this time.

 

[ttakacs]

Thank you. So the problem was something like to indicate time dilation in a very fast inertial spaceship by a mathematical pendulum - without gravity.. I will check the page with double relativity you suggested, thx.

By the way IF Planck length was the smallest length and Planck time was the smallest time could we say that our descriptions of nature with differential equations are not correct (even if very good approximations) since we cannot make limit values in derivatives e.g. dt and dx, etc. cannot go to zero?

Also I have a question related to quantification. In physics continuous fields are often used to describe nature. However mathematics might teach us to be careful with infinity: maybe the best example is the Banach-Tarski paradox that tells indirectly something that physical matter cannot be continuous since it could go against conservation of mass. Could you explain please how this is bypassed in physics?