Arbitrariness in Special Relativity

[planck42]

In an online lecture on Special Relativity, the instructor asserts that in the space-time coordinate system, $$(d{\tau})^{2}=(dt)^{2}-(dx)^{2}$$ with $$\tau$$ representing the proper time in a frame moving with velocity v, t representing a period of time measured from an inertial reference frame, and c being clearly treated as 1. It is known from the Lorentz transformations that x and t are related through hyperbolic angles, but surely this cannot be sufficient to assume that the proper time is as it is given. If there is no clear mathematical basis for $$(d{\tau})^{2}=(dt)^{2}-(dx)^{2}$$, then how can it be possible to derive such an equation willy-nilly?

 

[Dale]

That is the definition of proper time. You don't derive definitions.

 

[JesseM]

I think you can in a sense "derive" it from the fact that coordinate time in any frame is defined in terms of proper time on a network of clocks at rest in that frame at every possible position (synchronized according to the Einstein synchronization convention). If you know that the proper time interval between two events on the worldline of an inertial clock must match up with the coordinate time between those events in the frame where the clock is at rest (i.e. the frame where the two events have the same position coordinate), then the definition of proper time between these events in other frames follows from the Lorentz transformation.

 

[Dale]

Yes, you can see that simply by setting dx=0 in the above.

 

[yuiop]

Willy-nilly derivation

The Lorentz transformation of the time coordinate:

$$t = \tau \cosh(\phi) - x \sinh(\phi)$$

(where $\phi = \tanh^{-1}(v)$)

can be recast as:

$$t = \frac{\tau - vx}{\sqrt{1-v^2}}$$

From this relation it is very easy to obtain the time dilation factor:

$$\frac{dt}{d\tau} = \frac{1}{\sqrt{1-v^2}}$$

It follows that:

$$d\tau^2 = dt^2(1-v^2)$$

and since v = dx/dt,

$$d\tau^2 = dt^2 -dx^2$$

 

[planck42]

Clearly I didn't try hard enough to understand the concept.

 

[starthaus]

In an online lecture on Special Relativity, the instructor asserts that in the space-time coordinate system, $$(d{\tau})^{2}=(dt)^{2}-(dx)^{2}$$ with $$\tau$$ representing the proper time in a frame moving with velocity v, t representing a period of time measured from an inertial reference frame, and c being clearly treated as 1. It is known from the Lorentz transformations that x and t are related through hyperbolic angles, but surely this cannot be sufficient to assume that the proper time is as it is given. If there is no clear mathematical basis for $$(d{\tau})^{2}=(dt)^{2}-(dx)^{2}$$, then how can it be possible to derive such an equation willy-nilly?

Start with the invariance of the Lorentz metric:

$$(d{\tau})^{2}-(d\zeta)^{2}=(dt)^{2}-(dx)^{2}$$

In the proper frame , there is no motion, so $$d \zeta=0$$

You get the definition of proper time:

$$(d{\tau})^{2}-0=(dt)^{2}-(dx)^{2}$$