JDoolin

Gold Member

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## Main Question or Discussion Point

**"Locally Lorentz"**

Mister Thorne Wheeler, "Gravitation" asks "What does it mean to say that the geometry of a sufficiently limited region of spacetime in the real physical world is Lorentzian?"

The follow this up with two answers, neither of which appears to have much to do with the question. Instead, they give formulas to calculate proper time and proper distance. This barely scratches the surface of what it means to be Lorentz. Certainly, a body that takes an accelerated path through spacetime ages less than a body that takes an inertial path.

But what MTW fails to do is to get across a deeper undestanding of the paradox involved, and more particularly the resolution to this paradox. In fact, I can't find a single example in MTW where they demonstrate any competent expanation or deeper understanding of Special Relativity. They seem to start and end with the notion that Special Relativity is completely summed up by one equation:

[tex]s^2=-\tau^2=x^2+y^2+z^2-t^2[/tex]

I have no problem with having one equation to sum up Special Relativity, I just think the chose the wrong one. As for me, I don't think that SR is summed up by the calculation of the space-time-interval between events. Instead, it is summed up by the

**Lorentz Transformation Equations.**

Let me offer a single example of what the Lorentz Transformations

*do*. Then we can ask whether that operation represents a "global" or a "local" application of Lorentz, and whether it makes sense to say that physics is only "locally" Lorentz.

Here is the example:

I have a particle observer one foot away from a wall. Roughly 1 nanosecond ago, light came from the wall which is currently being observed by our observer.

The space-time coordinates of this observer is (ct, x) = (-1, 1); that is, 1 second ago, and one foot away.

Now, our observer undergoes a tremendous acceleration toward the wall of 0.99999999959c. (This corresponds to a change in rapidity of 10. tanh(10)=0.99999999959c.

The event (-1,1) is transformed by Lorentz Transformation as

[tex]\left(

\begin{array}{cc}

\cosh (\varphi ) & -\sinh (\varphi ) \\

-\sinh (\varphi ) & \cosh (\varphi )

\end{array}

\right)=

\left(

\begin{array}{c}

-22026 \\

22026

\end{array}

\right)

[/tex]

So this event, which happened only 1 foot away, now happened 22026 feet away; that is, over four miles away.

In general with these large numbers, (velocities extremely close to c) it is a simple calculation, once you know the rapidity. If you have a rapidity of 100, then the multiple is 2*cosh(100)=2.68*10

^{43}. With a rapidity change of 1000, the multiple is 2*cosh(1000)=1.97*10

^{434}feet (6.36*10

^{414}light years.)

So...

Is this exampe local? The proper time between these two events was zero. The proper distance between these two events is zero. This would seem to be about as "local" as you can get.

However, by applying the Lorentz Transformation to these two events, we were able to make them as far apart as we wanted in coordinate space and time. A google google google google times as big as the universe. However, the proper time, and proper distance between these events remains zero.

So how can anyone justify even using the phrase "locally Lorentz" If any physics is Lorentz at all, it can be stretched over arbitrarily large swaths of spacetime--not local at all.