Understanding Einstein's Theory: The Role of Spacetime in Motion

[Dorje]

In The Elegant Universe, Briane Greene states that, "Einstein proclaimed that all objects in the universe are always traveling through spacetime at one fixed speed - that of light." If something is sitting still, then all the object's "motion" is in the time dimension. Can someone demonstrate how this claim works out mathematically?

 

[cookiemonster]

The "distance" (it takes on a subtely different meaning for SR, i.e. it is spacetime distance) formula for SR is

$$ds^2 = dx^2 + dy^2 + dz^2 + (icdt)^2$$

If you fix the s and let the x, y, and z terms be insigificant, the time term is dominant. On the other hand, if the x, y, and/or z terms are significant, then the time term must be less significant.

cookiemonster

 

[da_willem]

I think Brian Greene explains it mathematically at the end of the book, in one of his 'footnotes'!

 

[DW]

Originally posted by cookiemonster
The "distance" (it takes on a subtely different meaning for SR, i.e. it is spacetime distance) formula for SR is

$$ds^2 = dx^2 + dy^2 + dz^2 + (icdt)^2$$

If you fix the s and let the x, y, and z terms be insigificant, the time term is dominant. On the other hand, if the x, y, and/or z terms are significant, then the time term must be less significant.

cookiemonster

The imaginary there is quite outdated. The source of the sign differennce is better understood when it is written in terms of the Pauli matrices including the 2x2 identity matrix $$\sigma _0$$:
$$(\sigma _{0}dc\tau)^2 = (\sigma _{0}dct)^2 + (\sigma _{x}dx)^2 + (\sigma _{y}dy)^2 + (\sigma _{z}dz)^2$$.
The Pauli matrices serve as a description of an orthonormal basis with the 2x2 identity associated with time. Boosts between frames are then described as rotatons in spacetime and this spacetime structure revealing spin half eigenvectors corresponding to rotations in space is then understood as the source of such quantum mechanical phenomenon. The ict concept is as outdated and uninstructive as the "relativistic mass" missnomer.

 

[DW]

Originally posted by Dorje
In The Elegant Universe, Briane Greene states that, "Einstein proclaimed that all objects in the universe are always traveling through spacetime at one fixed speed - that of light." If something is sitting still, then all the object's "motion" is in the time dimension. Can someone demonstrate how this claim works out mathematically?

Yes. Consider the velocity four-vector, $$U^\lambda = \frac{dx^\lambda}{d\tau}$$. The four dimensional speed that he is referring to is
$$|U| = [g_{\mu}_{\nu}U^{\mu}U^{\nu}]^{\frac{1}{2}} = c$$.
In the correct relativistic description which he is using everything always travels at this speed c in that way following geodesics when unacted on by real forces and all that real forces do is deflect them from geodesic motion. The real forces just rotate their direction in spacetime.

 

[pmb_phy]

Originally posted by Dorje
In The Elegant Universe, Briane Greene states that, "Einstein proclaimed that all objects in the universe are always traveling through spacetime at one fixed speed - that of light." If something is sitting still, then all the object's "motion" is in the time dimension. Can someone demonstrate how this claim works out mathematically?

That's a very poor way to think. This is not "traveling" in the normal sense. To "travel" means that one's position in space changes by the amount dr during a time interval dt and the speed of then said to be dr/dt. Greene is speaking not of a spatial displacement per time interval. He is referring to a spacetime interval, ds, in spacetime during a proper time interval dT. But this is another quite different concept that that of dr/dt.

Greene is not referring to a spatial displacement dr and he's not referring to a time interval dt. He's referring to a spacetime displacement ds and a proper time interval dT. But moving through spacetime with speed "c" Greene means that if U is the particle's 4-velocity. Defined as follows

Let dX be a space time displacement defined as

dX = (ct, dx, dy, dz)

Then the particle's 4-velocity U is defined as

U = dX/dT


Then it follows that U*U = c² or U*U = -c² depending on the choice of metric you choose. However if the particle is a photon then this relation is meaningless as is the 4-velocity as it is defined above. So don't get the idea that a photon's speed through spacetime is c.

It's highly unintuitive to say the all objects move through spacetime at speed c since the term "motion" brings to mind spatial movement. Even a body at rest has a non-zero value of ds. And when it comes to the speed of light then you can't even assign a speed to it in the sense that Greene is speaking.

 

[Dorje]

Thanks for the info! I think the problem with the quote does lie in the use of language -- the concepts of "traveling" and "motion" are colloquially associated with spatial displacement vs. time.

 

[turin]

That's a very poor way to think.

As much as I don't care for the expositions of Briannne Greeeennnne (I don't know how to spell it), I feel I have to throw my two cents into the pot to say that I think it is a very good way to think [about it], especially when considering GR.

I would just make one minor adjustment from "time dimension" to "time direction."