I Is Velocity Through Spacetime Always Equal to 'c'?

[PeterDonis]

if it is spacelike then movement in it, and specifically orbits, are not even possible, breaking the whole point of the analogy ;)

Not really. The point of the analogy is to provide a Newtonian sort of viewpoint, where things move in "space" with respect to "time", and the rubber sheet describes the "space" in which they move. It's not an invalid analogy, just limited and often misleading because people don't understand its limitations.

 

[PeterDonis]

I think what Prof Greene was referring to was a "speed through time" rather than spacetime.

Greene is describing what @SiennaTheGr8 and @kith were describing in posts #10 and #15. "Speed through spacetime" is constant, it's always $c$, but "speed through time" gets smaller as "speed through space" gets larger. So both "speed through spacetime" and "speed through time" appear in this viewpoint. You can't discard "speed through spacetime" because the fact that it is constant is what explains why "speed through time" decreases as "speed through space" increases.

 

[m4r35n357]

Greene is describing what @SiennaTheGr8 and @kith were describing in posts #10 and #15. "Speed through spacetime" is constant, it's always $c$, but "speed through time" gets smaller as "speed through space" gets larger. So both "speed through spacetime" and "speed through time" appear in this viewpoint. You can't discard "speed through spacetime" because the fact that it is constant is what explains why "speed through time" decreases as "speed through space" increases.

OK, after a bit more searching I found this thread where there is a quote from "An Elegant Universe". If you squint really hard he actually talks about both! I'll settle for that ;)

 

[PeterDonis]

after a bit more searching I found this thread where there is a quote from "An Elegant Universe".

Yes, I remember that thread well. My posts there pretty much sum up my opinion of Greene's pop science books.

 

[PeterDonis]

If you squint really hard he actually talks about both!

One reason why Greene might shy away from the term "speed through spacetime" is that, if you actually try to extend the analogy to photons the way he implies (i.e., for a photon, its "speed through space" is $c$ so its "speed through time" is zero), the analogy breaks down! A photon's "speed through spacetime", by the definition the analogy uses, is not $c$; it's zero (because photon worldlines have null tangent vectors).

Or, to put it another way, the equation $c^2 = v_{\text{time}}^2 + v_{\text{space}}^2$, which is valid for timelike objects (because it's just a rewriting of the interval equation, as @kith showed in post #15), is not valid for photons! So trying to argue from this analogy that "a photon's speed through time is zero" is, mathematically, not correct. Yet Greene's words and graphs strongly suggest such an argument. Emphasizing "speed through spacetime" might make this wrong implication too obvious.

 

[Matterwave]

One reason why Greene might shy away from the term "speed through spacetime" is that, if you actually try to extend the analogy to photons the way he implies (i.e., for a photon, its "speed through space" is $c$ so its "speed through time" is zero), the analogy breaks down! A photon's "speed through spacetime", by the definition the analogy uses, is not $c$; it's zero (because photon worldlines have null tangent vectors).

Or, to put it another way, the equation $c^2 = v_{\text{time}}^2 + v_{\text{space}}^2$, which is valid for timelike objects (because it's just a rewriting of the interval equation, as @kith showed in post #15), is not valid for photons! So trying to argue from this analogy that "a photon's speed through time is zero" is, mathematically, not correct. Yet Greene's words and graphs strongly suggest such an argument. Emphasizing "speed through spacetime" might make this wrong implication too obvious.

I raise this exact issue in my post #14. I am extremely dissatisfied with this analogy.

 

[Sorcerer]

I raise this exact issue in my post #14. I am extremely dissatisfied with this analogy.

What if it's explicitly mentioned that it doesn't apply to light?

 

[PeterDonis]

What if it's explicitly mentioned that it doesn't apply to light?

Is it? Does Greene ever say that?

 

[Matterwave]

What if it's explicitly mentioned that it doesn't apply to light?

You still run into my second objection. To me, the tangent vector is arbitrarily normalized to $c$...one could obtain the same physics by choosing a different normalization (and hence different affine parameterization) for worldlines. Using the proper time parameterization is nice, but from the viewpoint of GR, it's just a convenient parameterization to use - I would not elevate it to some physical principle about "speed through space time".

 

[Sorcerer]

Is it? Does Greene ever say that?

I can't remember. I haven't read his books in like three years, and my rabbit ate most of them for some reason (totally not joking about this).

 

[PAllen]

You still run into my second objection. To me, the tangent vector is arbitrarily normalized to $c$...one could obtain the same physics by choosing a different normalization (and hence different affine parameterization) for worldlines. Using the proper time parameterization is nice, but from the viewpoint of GR, it's just a convenient parameterization to use - I would not elevate it to some physical principle about "speed through space time".

Yes, in fact some authors always normalize 4 velocity to 1, even when not using c=1. An example is Peter Bergman, in his classic 1942 text.

 

[Sorcerer]

You still run into my second objection. To me, the tangent vector is arbitrarily normalized to $c$...one could obtain the same physics by choosing a different normalization (and hence different affine parameterization) for worldlines. Using the proper time parameterization is nice, but from the viewpoint of GR, it's just a convenient parameterization to use - I would not elevate it to some physical principle about "speed through space time".

Yeah, but in my humble opinion, as I learn more about relativity, it's so much more subtle than novices and lay people think that no matter what happens there are going to be horrid misunderstandings along the way. Someone here mentioned something about there being differences in difficulty of clearing up misunderstandings, depending on which misunderstanding, and I think that's valid, but no matter what happens people are likely going to come into it with awful, incorrect understandings and are likely going to keep them for a while.

For example, in first semester physics, I thought all special relativity was was taking old physics and slapping a gamma factor on everything- just because I saw it on the Galilean x-coordinate transformation and Newtonian momentum. So yeah, it's a mess no matter what. Just a matter of minimizing how badly students screw it up along the way, I guess. ;)

The physics educators will, of course, know which misunderstandings pop up the most and which linger the longest in the minds of students.

 

[vanhees71]

Yes, in fact some authors always normalize 4 velocity to 1, even when not using c=1. An example is Peter Bergman, in his classic 1942 text.

That's a very wise convention, which I also follow in my writeup on relativity here:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

The reason is that it is very convenient, because usually you need the four-velocity to project in a coordinate-independent way to time- and spatial components of other vectors in an instantaneous rest frame of the particle (or in fluid dynamics the local restframe of the fluid element).

 

[SiennaTheGr8]

Yes, in fact some authors always normalize 4 velocity to 1, even when not using c=1. An example is Peter Bergman, in his classic 1942 text.

This is how I like to do things. More generally, I always use $ct$ and $c \tau$ (never a $c$-less $t$ or $\tau$), including when differentiating with respect to coordinate or proper time. For example, I express coordinate acceleration $\vec \zeta$ as the second $ct$-derivative of position $\vec r$, so that it has the dimension of inverse length (i.e., $\vec \zeta = \vec a / c^2$). As for dynamics, it's always $E_0$ and $\vec p c$ for me; never $m$ or $\vec p$. Put it all together, and the only $c$'s are those attached to a $t$, $\tau$, or $\vec p$. So I don't set $c=1$, but I get most of the benefits of doing so.

 

[runningc]

To 'robphy'
I'm sorry, but when I try to access your attachment, 231822, a message comes up saying that it is not available. Can you help?
Runningc