What does this section mean from Gravitation by Misner, Thorne and Wheeler?

Question

What does it mean to say that the geometry of a sufficiently limited region of spacetime in the real physical world is Lorentzian?

A. Coordinate-free language (Robb 1936):

Let $\mathcal{AZ}$ be the world line of a free particle. Let $\mathcal{B}$ be an event not on this world line. Let a light ray from $\mathcal{B}$ strike $\mathcal{AZ}$ at the event $\mathcal{Q}$. Let a light ray take off from such an earlier event $\mathcal{P}$ along $\mathcal{AZ}$ that it reaches $\mathcal{B}$. Then the proper distance $s_\mathcal{AB}$ (spacelike separation) or proper time $\tau_\mathcal{AB}$ (timelike separation) is given by $$ s_\mathcal{AB}^2 \equiv - \tau_\mathcal{AB}^2 = - \tau_\mathcal{AQ}\tau_\mathcal{AP}.$$

The core of my question is why proper distance squared is equivalent to the negative of proper time squared?

The metric convention used in the book is (-+++).

From my understanding, the square of the proper time is proportional to the square of the invariant line element spacetime interval, ds^2. I don't know where the proper length comes into that though.

Answer 1

As @PaulT says, their notation is confusing.

For thoroughness, that passage should probably should say:
(in the (-+++)-signature convention) for a 4-vector displacement $\bf \vec s=\mathcal{AB}$, the square-interval is $$\Delta \bf s^2_\mathcal{AB}= (-\tau_\mathcal{AQ}\tau_\mathcal{AP}).$$

• When 4-vector $\bf \vec s=\mathcal{AB}$ is spacelike, then (in the (-+++)-signature convention) $\bf \Delta s^2_\mathcal{AB}>0$, and the proper-distance $s_\mathcal{AB}$ along that spacelike-displacement is defined and is determined by the radar measurement $$s_\mathcal{AB}^2 \equiv +\Delta \bf s^2_\mathcal{AB} = (-\tau_\mathcal{AQ}\tau_\mathcal{AP})>0,$$ with chronological ordering $\mathcal P < \mathcal A < \mathcal Q$ along $\mathcal{AZ}$ (so that $\tau_{\mathcal{AQ}}>0$ and $\tau_{\mathcal{AP}}<0$).
• When 4-vector $\bf \vec s=\mathcal{AB}$ is timelike, then (in the (-+++)-signature convention) $\bf \Delta s^2_\mathcal{AB}<0$, and the proper-time $\tau_\mathcal{AB}$ along that timelike-displacement is defined and is determined by the radar measurement $$\tau_\mathcal{AB}^2 \equiv -\Delta \bf s^2_\mathcal{AB}= -(-\tau_\mathcal{AQ}\tau_\mathcal{AP})>0$$ with chronological ordering $\mathcal A < \mathcal P < \mathcal Q$ (so that $\tau_{\mathcal{AQ}}>0$ and $\tau_{\mathcal{AP}}>0$).
• When 4-vector $\bf \vec s=\mathcal{AB}$ is lightlike, then $\bf \Delta s^2_\mathcal{AB}=0$
  with either $\mathcal A=\mathcal P\leq \mathcal Q$ (for $\mathcal{AB}$ for future-lightlike) or $\mathcal P\leq \mathcal Q=\mathcal A$ (for $\mathcal{AB}$ for past-lightlike) along $\mathcal{AZ}$.

On the next page, they show that, in the $\mathcal{AZ}$-frame with conveniently chosen axes that
$\tau_\mathcal{AP}=\Delta t_{\mathcal{AB}}-\Delta x_{\mathcal{AB}}$ and
$\tau_\mathcal{AQ}=\Delta t_{\mathcal{AB}}+\Delta x_{\mathcal{AB}}$
so that $\tau_\mathcal{AQ}\tau_\mathcal{AP}=\Delta t_{\mathcal{AB}}^2 - \Delta x_{\mathcal{AB}}^2$ and thus, we see the signature with $$(-\tau_\mathcal{AQ}\tau_\mathcal{AP})= -\Delta t_{\mathcal{AB}}^2 + \Delta x_{\mathcal{AB}}^2.$$

Answer 2

I find that definition very confusing. Calculationally, that statement is correct, but it muddles the conceptual understanding of what is going on.

The proper spacetime separation between two events in spacetime, A and B, is $$\Delta s^2 = \Delta\vec{s}\cdot\Delta\vec{s},$$ where $\Delta \vec{s} = \vec{s}_A - \vec{s}_B$, and $\vec{s}_A\rightarrow(ct_A, x_A, y_A, z_A)$ is the spacetime 4-vector position of event A. This separation can be timelike, spacelike, or lightlike (null).

The proper time is only defined for timelike separated events. For these types of events $\Delta s^2$ is negative and proper time, $\tau = \sqrt{-\Delta s^2}$. The proper time is the time interval recorded by a co-moving observer. For a co-moving observer the two events, A and B, happen at the same place. There is no spatial separation of the two events according to the co-moving observer.

The proper length is only defined for spacelike separated events. For these types of events $\Delta s^2$ is positive and proper length, $\ell = \sqrt{\Delta s^2}$. The proper length is the spatial distance between the events according to an observer who observes the events to be simultaneous. For this observer the two events, A and B, happen at the same time. There is no temporal separation between the two events according to this observer.

Two events cannot have both proper length and proper time, because the events cannot be both spacelike and timelike separated.

Lightlike or null separated events have $\Delta\vec{s}^2=0$. It is not possible to construct a co-moving reference frame or find a reference frame where the events occurred simultaneously, so these events have not proper time or proper length.