Does everything move at the speed of light?

[Max364]

TL;DR Summary

Came across this video: https://www.youtube.com/watch?v=TJmgKdc7H34
where SR, time dilation, twin paradox, 4D etc are all brilliantly explained with simple graphical illustrations.

...but is it correct to say: Speed² (in space) + Time² (in space) = 1 ?? derived using Pythagorus (explained at 10-13 minutes in)....the trade off between Speed and Time, at maximum speed Time is zero and at max Time, Speed would be zero.

 

[PeroK]

...but is it correct to say: Speed² (in space) + Time² (in space) = 1 ?? derived using Pythagorus (explained at 10-13 minutes in)....the trade off between Speed and Time, at maximum speed Time is zero and at max Time, Speed would be zero.

You can calculate easily enough that for any timelike path (one that any particle with mass follows), the magnitude of the particle's four-velocity is $\pm c^2$. See here for example:

https://en.wikipedia.org/wiki/Four-velocity#Magnitude

There is no reason, however, to interpret that as moving through spacetime at the speed of light. The speed of light is a measure of motion through space with respect to time. "Motion" through spacetime is not really well-defined.

Note also that light moves along null or light-like paths that have zero spacetime distance - which is a very important point in the theory of relativity.

 

[Ibix]

Sort of. It's not a very helpful view, though - it leads to wildly unhelpful notions of paths through spacetime and it's a dead-end if you want to start to understand general relativity.

Using four-velocity properly, rather than mangling it by components as the video does is a much more fruitful approach.

 

[Mister T]

The entire argument presents an analogy to the actual physics. Analogies can be useful in understanding some physics (that's the reason for creating them) but they can always be taken too far and lead to erroneous conclusions. In other words, they are of limited utility.

Epstein's book is a good example.

First of all I know what it means to travel through space, and in particular to do so at the speed of light. Determining the speed involves a calculation using both space and time separately.

I don't know what it means to travel through spacetime. The very phrase is a meaningless word salad to me.

The passage of time for an object moving at light speed Is undefined (it has no meaning). To say it has a value of zero is to give meaning to it, which is something that physics cannot do for you.

While I agree that Epstein's book may have some use, in 33 years of answering questions about it I conclude that it often leads the reader to logical contradictions and hence can be a source of confusion.

 

[Orodruin]

Speed² (in space) + Time² (in space) = 1 ??

This is a horrible way to view 4-velocity normalisation. It is in contrast to basically every proper coverage in that they have defined speed in space as $dx/dt$ with $x$ and $t$ being coordinates and the "time" as the proper time $s$ of the object. The more appropriate form of the equation would use proper speed in time and space, i.e., coordinate change per proper time rather than coordinate change per coordinate time and be
$$
\left(\frac{dt}{ds}\right)^2 - \left(\frac{dx}{ds}\right)^2 = 1,
$$
i.e., using the Minkowski metric.

The presented form has "speed in space" $= dx/dt$ and "speed in time" = $ds/dt$, which is confusing but obtainable from the appropriate form as
$$
\left(\frac{dt}{ds}\right)^2 - \left(\frac{dx}{ds}\right)^2 = \left(\frac{dt}{ds}\right)^2\left[ 1 - \left(\frac{dx}{dt}\right)^2\right]
$$
and
$$
1 = \left(\frac{dt}{dt}\right)^2 = \left(\frac{dt}{ds}\right)^2 \left(\frac{ds}{dt}\right)^2
$$
leading to
$$
1 - \left(\frac{dx}{dt}\right)^2 = \left(\frac{ds}{dt}\right)^2
$$
and therefore
$$
1 = \left(\frac{ds}{dt}\right)^2 + \left(\frac{dx}{dt}\right)^2
$$

 

[robphy]

An old post of mine on this topic:
https://www.physicsforums.com/threa...ning-light-speed-travelers.75111/#post-565681