## Main Question or Discussion Point

I tell them: (first, explain spacetime)

"Relative to anyone else, we move through space-time at the speed of light.

Take you and I standing here, you are not moving in space relative me so I see you moving through time at the speed of light. If somebody flew past at near the speed of light, I wouldn't see them moving through time very much at all.

Since space velocity and time velocity always combine to the same total, people who are moving through space relative to you, move slower through time relative to you."

In my opinion, a scientific theory shouldbe as sensible and accessible as possible to everyone. I dislike books that play up SR as some mystical time space weirdness. If any one has a better way to explain SR, or a way to explain length contraction in a nutshell, I would like to hear it.

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dextercioby
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Daniel.

JesseM
Crosson said:
I tell them: (first, explain spacetime)

"Relative to anyone else, we move through space-time at the speed of light.

Take you and I standing here, you are not moving in space relative me so I see you moving through time at the speed of light. If somebody flew past at near the speed of light, I wouldn't see them moving through time very much at all.

Since space velocity and time velocity always combine to the same total, people who are moving through space relative to you, move slower through time relative to you."
This idea of explaining relativity in terms of "moving through spacetime at the speed of light" is not a very good starting place to think about relativity, in my opinion--it's certainly not how relativity is described in most textbooks, and I'm pretty sure it's not how Einstein ever described it either. In fact, Brian Greene is the only author I have seen who describes relativity this way, and he explains the justification in an endnote in The Elegant Universe (p. 392):
For the mathematically inclined reader, we note that from the spacetime position 4-vector $$x = (ct, x_1, x_2, x_3) = (ct, \vec{x})$$ we can produce the velocity 4-vector $$u$$, $$dx/d\tau$$, where $$\tau$$ is the proper time defined by $$d\tau^2 = dt^2 - c^{-2}(dx_1^2 + dx_2^2 + dx_3^2)$$. Then, the "speed through spacetime" is the magnitude of the 4-vector $$u$$, $$\sqrt{((c^2dt^2 - d\vec{x}^2)/(dt^2 - c^{-2}d\vec{x}^2))}$$, which is identically the speed of light, $$c$$. Now, we can rearrange the equation $$c^2(dt/d\tau)^2 - (d\vec{x}/d\tau)^2 = c^2$$, to be $$c^2(d\tau/dt)^2 + (d\vec{x}/dt)^2 = c^2$$. This shows that an increase in the object's speed through space, $$\sqrt{(d\vec{x}/dt)^2}$$, must be accompanied by a decrease in $$d\tau/dt$$, the latter being the object's speed through time (the rate at which time elapses on its own clock, $$d\tau$$, as compared with that on our stationary clock, $$dt$$).
This is really just a sort of mathematical trick, and although the vector $$u$$ above is referred to as the "4-velocity", I'm not aware of anyone but Greene who calls its magnitude "speed through spacetime", and I don't think it really makes much sense to describe it this way--if "speed" is ordinarily the rate your position in 3D space is changing with time, would "speed through spacetime" mean the rate an object's position in 4D spacetime is changing as a function of time? The "distance" between two points in spacetime is given by $$\sqrt{dt^2 - c^{-2}(dx_1^2 + dx_2^2 + dx_3^2)}$$, so if in your reference frame an object moves a distance $$\sqrt{dx_1^2 + dx_2^2 + dx_3^2}$$ in time $$dt$$, then it seems that we should say the "distance through spacetime" it has travelled is $$d\tau = \sqrt{dt^2 - c^{-2}(dx_1^2 + dx_2^2 + dx_3^2)}$$, therefore its "speed through spacetime" should be $$d\tau/dt$$, and yet Greene defines this as the "speed through time". Of course, since "speed through time" and "speed through spacetime" are not standard terms in relativity, Greene is in some sense free to pick any definitions he likes, but as I said all this seems unecessarily confusing and doesn't really tell you about the fundamental basis of relativity.

The real fundamental basis of relativity is the following two postulates made by Einstein:

1. All observers moving inertially (ie not accelerating) will observe the laws of physics to work the same way in their own reference frame

2. Maxwell's laws hold exactly in at least one inertial reference frame (at the classical level, anyway), which by postulate 1 means they hold exactly in every inertial reference frame; since Maxwell's laws predict that the speed of electromagnetic waves is always c, the speed of light must be c in every inertial reference frame.

(note--before relativity, most physicists believed that Maxwell's laws only held exactly in a single inertial reference frame, the rest frame of the 'luminiferous aether' which was imagined to be a medium filling all of space, with light as a soundwave in this medium)

From the idea that the speed of light is c in all inertial reference frames, and the fact that the laws of physics should be defined by the same equations in every reference frame, you can derive every other idea in relativity, like the idea that each observer will see moving clocks slow down in his frame, and moving rulers shrink.

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Heh. I gotta admit, I'm curious as to what Crosson's definition of a "layman" is. People skim through The Elegant Universe once, and within a week they're referring to other people as "laymen". I wish that term had never been adopted by the learn'ed community. It sounds rediculous. :yuck:

Perhaps my post seemed condescending (use of the word Laymen) but it is very rude of you to accuse me of reading "the elegant universe".

You people rush in to give me your out-of-the-can relativity lecture, I don't need that. I asked if anyone had creative, original ways of expressing relativity to people who don't study physics! I shared my "relativity on a T-shirt" idea (which I apparently share with crackpot Brian Greene), and you guys can only attack me personally.

Yes JesseM, that is the very same "mathematical trick" which justifies my interpretation, since all I am doing is giving (non-physics) people a qualitative pneumonic analogy.

In my opinion, you people are viscious and insecure, as if recommending me "special relativity for school children" some how makes you seem smarter. If all you can do is recite einstein's 1905 paper ad verbatim, then that is a hobby for yourself that no one else will care about. Only if you can have original thoughts to further our understanding, then you are worth something as a physicist. So spend your time reading those texts that you think you understand, and make sure to insult and discourage anyone who tries to do some thing creative and inclusive.

I wouldn't be surprised if you are all to embarassed to respond, as you should be.

JesseM
Crosson said:
Perhaps my post seemed condescending (use of the word Laymen) but it is very rude of you to accuse me of reading "the elegant universe".

You people rush in to give me your out-of-the-can relativity lecture, I don't need that. I asked if anyone had creative, original ways of expressing relativity to people who don't study physics! I shared my "relativity on a T-shirt" idea (which I apparently share with crackpot Brian Greene), and you guys can only attack me personally.
No one said Greene was a crackpot, of course--his math is correct, I just don't think it makes intuitive sense to label the magnitude of the 4-velocity "speed through spacetime", although since the term has no preexisting meaning this is a matter of taste (I'm not even sure if it makes much sense to think of the 4-velocity as an actual 'velocity', despite the name, but I'd have to think about that one a little more). If you think this label is intuitive, could you address my criticism above about why it would make more sense to label $$d\tau / dt$$ the "speed through spacetime"? If "speed" is normally distance/time, isn't it more natural to let (distance in spacetime between two events)/(time between two events) be the speed through spacetime?
Crosson said:
If all you can do is recite einstein's 1905 paper ad verbatim, then that is a hobby for yourself that no one else will care about. Only if you can have original thoughts to further our understanding, then you are worth something as a physicist. So spend your time reading those texts that you think you understand, and make sure to insult and discourage anyone who tries to do some thing creative and inclusive.
Unlike other posters on this thread I didn't question your own knowledge of relativity, I just questioned the pedagogical value of your way of explaining relativity to a layman. I haven't even read Einstein's paper, but every text I have seen starts out with those two basic assumptions, probably because this approach makes it easiest to understand the motivation for other, weirder concepts in relativity like time dilation. For example, based on the idea that light must always travel at c in every frame, even without any math you can show why different observers must synchronize their clocks differently--just imagine a lightflash emitted at the midpoint of a rocket which is moving in my frame, you can see that in my frame the front of the rocket is moving away from the flash and the back is moving towards it so the light will arrive at the back first, but in the rocket's frame the light must hit both sides at the same time. Likewise, you can imagine an observer on a moving train shining a light straight down at the floor of the train--in my frame the light travels a diagonal path since the train is moving, and this path is longer than the path straight down which the observer on the train sees, so in order for distance/time to come out to c in both cases, his clocks must be running slow from my perspective. So there's a lot of relativity that can be understood intuitively in terms of the idea that everyone must get the same value for the speed of light, without the need for any mathematical derivations--I think this tried-and-true method is the best way of explaining relativity to beginners.

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chroot
Staff Emeritus
Gold Member
Crosson,

By identifying yourself as someone to whom hordes of laypeople come begging for enlightenment, you are essentially identifying yourself as a professional physicist.

The problem with that identification is that, to date, only one professional physicist has ever used your approach to educate laypeople. You therefore inadvertently identified yourself as either Brian Greene himself, or someone who read his book. Since most professional physicists don't go around using paperback-book pedagogy to actively teach people, you've also simultaneously identified yourself as a layperson. This contradictory identification makes many people cringe, as you've seen here.

The internet is absolutely full of people with deep delusions of grandeur who want nothing more than to be the personal, sole successor of Einstein himself. These people collectively probably know less relatively than your average upperclassman on the way to a BS in physics, however, so sneers are a common knee-jerk reaction to those anonymous netizens who take a holier-than-thou stance about their credentials.

You're new here -- stick around, and learn the lay of the land. We have a large number of professional educators here, and quite a few graduate students, too. What are your credentials?

- Warren

chroot said:
Crosson,

By identifying yourself as someone to whom hordes of laypeople come begging for enlightenment, you are essentially identifying yourself as a professional physicist.

The problem with that identification is that, to date, only one professional physicist has ever used your approach to educate laypeople. You therefore inadvertently identified yourself as either Brian Greene himself, or someone who read his book. Since most professional physicists don't go around using paperback-book pedagogy to actively teach people, you've also simultaneously identified yourself as a layperson. This contradictory identification makes many people cringe, as you've seen here.

The internet is absolutely full of people with deep delusions of grandeur who want nothing more than to be the personal, sole successor of Einstein himself. These people collectively probably know less relatively than your average upperclassman on the way to a BS in physics, however, so sneers are a common knee-jerk reaction to those anonymous netizens who take a holier-than-thou stance about their credentials.

You're new here -- stick around, and learn the lay of the land. We have a large number of professional educators here, and quite a few graduate students, too. What are your credentials?

- Warren

I couldn't have said it better.

BTW, I do not think Brian Greene is a crackpot. I merely think it is important to make a distinction between folks who think they are in the know and people who really are. If you do not have a Ph.D. in physics, you are a "layperson". There are no two ways about it. I do not have a Ph.D., therefore I am a layperson. For some reason, some people can't wrap their noodle around this idea.

I am sorry that I posted some rehash of Brian Greene's idea, it is certainly not a big enough deal for me to spend any time convincing you that I thought of it independently.

I still think that the "everyone moves through spacetime at the speed of light" is a concise, elegant and qualitatively accurate interpretation of special relativity that is at least allowed, if not suggested, by the mathematics. Compared to explaining einsteins postulates (the consequences of which are simple to work out with paper, but not in a casual conversation) my interpretation cuts to the chase.

I apologize for ever using the word laymen. My definition of a laymen, in this case, is someone who has only ever heard of Einstein, and does not know the content of his theories. This person, relative to myself is a layperson because I am:

Credentials: Undergraduate physics major, I just turned 19 and will be completing the typical undergraduate physics curriculum at the end of the spring semester. I have been thinking about SR for at least a decade (encyclopedia) and have developed an (original to me) derivation of the gamma factor which I believe is more straightfoward than the standard light clock. Currently my favorite subject is relativistic electrodynamics, and I think I have read just about every word on special relativity this side of the quantum limit.

That was exhaustive and immodest, but I am new here and chroot asked for it. I also listed all that so that I can ask the question:

Am I still a layman by your definition PBRMEASAP? I have met more than one Phd who didn't know his way around relativity conceptually, so I object to your definition. Althought you only assert a Phd as a neccesary (not sufficient) condition for not being a layman.

JesseM
Crosson said:
I still think that the "everyone moves through spacetime at the speed of light" is a concise, elegant and qualitatively accurate interpretation of special relativity that is at least allowed, if not suggested, by the mathematics. Compared to explaining einsteins postulates (the consequences of which are simple to work out with paper, but not in a casual conversation) my interpretation cuts to the chase.
Can you address my question about why "speed through spacetime" is not simply defined as (distance in spacetime)/time?

Crossen - you can expect rude treatment on this board by many of the posters - there are some good guys that are very patient and polite ... persons that are not threatened by some different ideas that may involve alternative interpretation of SR. If they were really up on their reading they would know that Brian Green was not the only well known author to have described relativity in this way (Both Hawking and Epstein have books that are aimed at getting across the ideas of SR to persons who want a simple analogy and both have used this metaphore - and for all we know it may be more than a metaphore since you can use that very simple concept to derive the Lorentz velocity transforms (the Gamma factor) which incidentally is the only aspect of SR that has been verified by experiment. Now watch them jump on me as always.

JesseM
yogi said:
Both Hawking and Epstein have books that are aimed at getting across the ideas of SR to persons who want a simple analogy and both have used this metaphore - and for all we know it may be more than a metaphore since you can use that very simple concept to derive the Lorentz velocity transforms (the Gamma factor) which incidentally is the only aspect of SR that has been verified by experiment. Now watch them jump on me as always.
Which books by Hawking and Epstein describe relativity this way? I'd like to know the actual quotes. Also, why are you calling it a metaphor? Greene's mathematical demonstration is perfectly sound, it's just a matter of aesthetic preference whether you think it makes sense to label the magnitude of the 4-velocity the "speed through spacetime" and $$d\tau / dt$$ the "speed through time". And when you say "you can use that very simple concept to derive the Lorentz velocity transforms", do you mean using the equation $$c^2(d\tau/dt)^2 + (d\vec{x}/dt)^2 = c^2$$ to derive $$1 / \sqrt{1 - v^2/c^2}$$, or do you mean using some simpler concept that increasing speed through time means decreasing speed through space and vice versa?

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Garth
Gold Member
JesseM said:
Which books by Hawking and Epstein describe relativity this way? I'd like to know the actual quotes. Also, why are you calling it a metaphor? Greene's mathematical demonstration is perfectly sound, it's just a matter of aesthetic preference whether you think it makes sense to label the magnitude of the 4-velocity the "speed through spacetime" and $$d\tau / dt$$ the "speed through time". And when you say "you can use that very simple concept to derive the Lorentz velocity transforms", do you mean using the equation $$c^2(d\tau/dt)^2 + (d\vec{x}/dt)^2 = c^2$$ to derive $$1 / \sqrt{1 - v^2/c^2}$$, or do you mean using some simpler concept that increasing speed through time means decreasing speed through space and vice versa?
I'd like to open myself to invective and join Crosson as one known to be happy to talk about "speed through space-time at c" (although I'd use velocity through space-time if I'm being precise), if those words are defined carefully.

The term "four-velocity" (4-velocity, velocity 4-vector etc.) is an absolutely standard and essential part of any textbook on relativity: Wald, Misner, Thorne & Wheeler, Weinberg, d'Inverno etc. etc. Using the term 'velocity' itsead of 'speed' is more rigourous and emphasises the need to resolve it into orthogonal coordinates.

However I would define it the other way round; $$dt/d\tau$$. One way to explain SR is to define $$\tau$$ by extending Pythagoras to form the SR time-like metric and then take classical (pre-SR) concepts such as momentum and replace t with $$\tau$$. Thus velocity in the x direction through 3D space $$dx/dt$$ becomes $$dx/d\tau$$. The norm of the four-velocity is always c and so the four-velocity of an object stationary in the observer's frame of reference is simply $$dt/d\tau = c$$. If 'speed thtough space-time' is now carefully defined to be 4-velocity and the word 'move' generalised into 4D then Crosson is right, and time dilation readily explained.

Be gentle with me!!

Garth

Crosson said:
Credentials: Undergraduate physics major, I just turned 19 and will be completing the typical undergraduate physics curriculum at the end of the spring semester. I have been thinking about SR for at least a decade (encyclopedia) and have developed an (original to me) derivation of the gamma factor which I believe is more straightfoward than the standard light clock. Currently my favorite subject is relativistic electrodynamics, and I think I have read just about every word on special relativity this side of the quantum limit.
Congratulations!

Crosson said:
Am I still a layman by your definition PBRMEASAP?
Yes. But I am willing to agree to disagree.

ohwilleke
Gold Member
The easiest way to explain special relativity to a layperson is not with mathematics or from first principals (i.e. the axioms that lead to special relativity). I think it is much easier to start by discussing its effects.

In other words: "When objects are moving close to the speed of light, relative to each other, they don't behave as you would expect them to behave. For example, (1) the closer you get to the speed of light (which is fixed relative to you no matter what speed or direction you are moving in), the harder it is to get there, and you can never go faster than the speed of light, and (2) time flows slowly for people who are moving rapidly relative to you, compared to the rate at which time moves for you. Time moves at a different rate for different objects."

This isn't complete, but gives a flavor of the important conclusions of the theory.

Crosson said:
I have developed an (original to me) derivation of the gamma factor which I believe is more straightfoward than the standard light clock.
Standard light clock is pretty simple, so it must be slick.
Still uses Pythagoras I assume.
Simple enough to share basic's here, or have a link to it?
Someone here can a least let you know if like something else used before.

JesseM
Garth said:
The term "four-velocity" (4-velocity, velocity 4-vector etc.) is an absolutely standard and essential part of any textbook on relativity: Wald, Misner, Thorne & Wheeler, Weinberg, d'Inverno etc. etc. Using the term 'velocity' itsead of 'speed' is more rigourous and emphasises the need to resolve it into orthogonal coordinates.
Sure, and I did mention earlier that 4-velocity is an accepted term, my point was that it's just a name, you shouldn't take the analogy with ordinary velocity too literally. I suppose it was given this name because of the similarity between the equation for ordinary velocity, $$v = (dx/dt, dy/dt, dz/dt)$$ and the equation $$u = (cdt/d\tau, dx/d\tau, dy/d\tau, dz/d\tau)$$, and also because the 4-velocity is always tangent to the worldline in the same way that ordinary velocity is tangent to the path through space. But what I'm saying is that the analogy breaks down when you compare the magnitude of 4-velocity to the magnitude of ordinary velocity--while ordinary velocity is distance/time, 4-velocity's magnitude is not (spacetime distance)/time (although you could define a new vector which would be parallel to 4-velocity but would have this magnitude).
Garth said:
However I would define it the other way round; $$dt/d\tau$$.
Define what the other way around? Speed through spacetime, or speed through time?
Garth said:
One way to explain SR is to define $$\tau$$ by extending Pythagoras to form the SR time-like metric and then take classical (pre-SR) concepts such as momentum and replace t with $$\tau$$. Thus velocity in the x direction through 3D space $$dx/dt$$ becomes $$dx/d\tau$$. The norm of the four-velocity is always c and so the four-velocity of an object stationary in the observer's frame of reference is simply $$dt/d\tau = c$$.
That should actually be $$cdt/d\tau = c$$, remember that the 4-velocity of a stationary object would be $$u = (cdt/d\tau, 0, 0, 0)$$. And the magnitude of the 4-velocity is always c regardless of whether the object is stationary in your reference frame or not.
Garth said:
If 'speed thtough space-time' is now carefully defined to be 4-velocity and the word 'move' generalised into 4D then Crosson is right, and time dilation readily explained.
But my point is that this is not a natural way to generalize "movement" into 4D, the natural way would be to look at the spacetime distance covered in a given amount of time. It seems that you are only calling the magnitude of the 4-velocity "speed through spacetime" because of the historical accident that $$u = (cdt/d\tau, dx/d\tau, dy/d\tau, dz/d\tau)$$ was given the name "4-velocity", not because it actually makes sense in terms of an analogy with our ordinary definition of speed.

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hey,

I just finished reading the article that dextercioby gave the link to, the one about speacial relativity (the two indian sisters). It was indeed enlightning, mainly because it will probably help me describe SR to non-physics people with more ease.

although I spent enough time learning about relativity in my physics courses, I always liked to understand things on the most basic level and that's why i have few questions for you: in that article, one sister describes how her watch will read a longer time for an event that takes place in the moving train, where her sister is sitting and reading a shorter time for the same even (the light beam) thus concluding that the first sister's watch is running slower, or that the passenger's watch is running faster. Now in reality, what is happening with those two watches? is there anything happening to the moving part inside the passenger's watch that is making it run faster? since the two watches are identical and powered by the same battery, is the high velocity at which the train is moving affecting the mechanical (or digital) functioning of the passenger's watch?

I do agree with SR. I even know of an experiment where they took an atomic clock and flew it around in an airplane and it showed a difference in time that an identical clock on the ground.

I'd appreciate your help...thanks a lot.

Moneer81: "I do agree with SR" Reminds of what my one of my history professors once said: "90% of the American people don't know what the Monroe Doctrine is, but they are for it."

Answering you questions JesseeM ...Epstein - Relativity Visualized at page 78 and 76 "The reason you can't go Faster than the Speed of Light is that you can't go slower - Everything including you is always moving at the speed of light."

Hawking - don't have the reference in front of me - I think it was "Search for a theory of Everything" My library is scattered between three different houses - if you have that reference it is on the page where the diagram shows the at rest girl progressing in time along the vertical axis and then vering off in the space-time direction as she moves relative to her at rest reference frame.

Hey chroot.....

Excellent post.....I especially liked this part......

chroot said:
The internet is absolutely full of people with deep delusions of grandeur who want nothing more than to be the personal, sole successor of Einstein himself. These people collectively probably know less relatively than your average upperclassman on the way to a BS in physics, however, so sneers are a common knee-jerk reaction to those anonymous netizens who take a holier-than-thou stance about their credentials.

- Warren
Quite true....back when I was an undergrad we had tons of students who thought they knew more, were more creative, were more intelligent, were more "visionary", and were more [insert some term describing how awesome they think they are] than the professors teaching the class, the professionals working in the field, and the tutors trying to help them.

Unfortunately.....learning and college in general does not always have the humbling effect that it should on people.

Russ

JesseM
yogi said:
Answering you questions JesseeM ...Epstein - Relativity Visualized at page 78 and 76 "The reason you can't go Faster than the Speed of Light is that you can't go slower - Everything including you is always moving at the speed of light."

Hawking - don't have the reference in front of me - I think it was "Search for a theory of Everything" My library is scattered between three different houses - if you have that reference it is on the page where the diagram shows the at rest girl progressing in time along the vertical axis and then vering off in the space-time direction as she moves relative to her at rest reference frame.
Thanks yogi. I'll look up what is said in that book, but keep in mind http://www.hawking.org.uk/info/news.html [Broken] from Hawking's website:
It has come to our attention that the book "The Theory of Everything: The Origin and Fate of the Universe" has been published. Professor Hawking would like to make it clear that he has not endorsed this book. The text was written by him many years ago, however the material has already been published in books such as 'A Brief History of Time'. A complaint was made to the Federal Trade Commission in the US in the hope that they would prevent the publication. We would urge you not to purchase this book in the belief that Professor Hawking was involved in its creation.
Since the text was written by him, it might well be that he would still consider this a good way of explaining relativity, although I don't think he used this explanation in any of the texts he actually chose to publish.

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JessieM ...so much of what is attributed to Hawking must be to some extent interpreted - seems it was Kitty Fergessen that took notes while interviewing him and later assembled them - The book I have has been in print a long time - don't know if its the same one that is being referred to in his web site. I will have a chance to look for it tomorrow.

Yogi

Garth
Gold Member
JesseM said:
Sure, and I did mention earlier that 4-velocity is an accepted term, my point was that it's just a name, you shouldn't take the analogy with ordinary velocity too literally. I suppose it was given this name because of the similarity between the equation for ordinary velocity, $$v = (dx/dt, dy/dt, dz/dt)$$ and the equation $$u = (cdt/d\tau, dx/d\tau, dy/d\tau, dz/d\tau)$$, and also because the 4-velocity is always tangent to the worldline in the same way that ordinary velocity is tangent to the path through space.
My point exactly, but to progress from a classical perspective of 3D + time to a SR 4D perspective one subsumes t into the space-time continuum and rates are measured wrt $$\tau$$.
JesseM said:
But what I'm saying is that the analogy breaks down when you compare the magnitude of 4-velocity to the magnitude of ordinary velocity
In the 'block universe' of a space-time diagram 'movement' takes on a different meaning and that has to be explained carefully. But the magnitude of 4-velocity differs from 3-velocity only in it always has the norm of c, unity in sensible units, which for a stationary object leaves it "'moving' through time at the speed of light" which was Crosson's point in the first place.
JesseM said:
Speed through spacetime, or speed through time?
Answer: Rate of change of x, y, z, or t with respect to $$\tau$$.
JesseM said:
That should actually be $$cdt/d\tau = c$$,
True, thank you - my mistake, I am so used to working in natural space-time (geometric) units where c = 1. so that
$$cdt/d\tau = c$$ -> $$dt/d\tau = 1$$ is light velocity.
JesseM said:
And the magnitude of the 4-velocity is always c regardless of whether the object is stationary in your reference frame or not. But my point is that this is not a natural way to generalize "movement" into 4D, the natural way would be to look at the spacetime distance covered in a given amount of time. It seems that you are only calling the magnitude of the 4-velocity "speed through spacetime" because of the historical accident that $$u = (cdt/d\tau, dx/d\tau, dy/d\tau, dz/d\tau)$$ was given the name "4-velocity", not because it actually makes sense in terms of an analogy with our ordinary definition of speed.
That is by your definition of speed in the classical 3D + time perspective. I choose to (carefully) define 4-velocity in a fully SR 4D perspective with time and space as equivalent dimensions (c = 1) and rates measured wrt $$\tau$$ as above.

Garth

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JesseM
Garth said:
That is by your definition of speed in the classical 3D + time perspective. I choose to (carefully) define 4-velocity in a fully SR 4D perspective with time and space as equivalent dimensions (c = 1) and rates measured wrt $$\tau$$ as above.
I'm fine with treating space and time as equivalent dimensions (although to make the notion of spacetime distance work out, it's better to make time an imaginary spatial dimension, treating 1 light-second as equivalent to i meters). But I still don't see how the magnitude of the 4-velocity is in any way analogous to the magnitude of ordinary velocity, since there seems to be nothing remotely analogous to distance/time there. You say that "rates" should be measured wrt $$\tau$$, but rates of what, exactly? Is there anything analogous to "distance" such that the magnitude of the 4-velocity can be considered to be equal to (something analogous to distance)/$$\tau$$?

Garth
Gold Member
rates of what, exactly?
Proper time $$\tau$$ is that time measured in the rest frame; moving and/or distant (in a gravitational field) clocks will run at a different rate, rulers will contract etc. Therefore measure the 'rate' of $$dt/d\tau, dx/d\tau$$ etc.

BTW Although time bears the same mathematical relationship to the space dimensions as the imaginary numbers do to the real, calling time "imaginary" ict has its own problems as MTW point out in ‘Gravitation’ page 51.

Garth

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