Measuring Time & Spatial Distances: Timelike vs Spacelike

[Kashmir]

My book writes
" The geometric distinction between timelike and spacelike distances is mirrored in the devices used to measure them. A clock is a device that measures timelike distances; a ruler is a device for measuring spacelike ones. Two nearby points or a timelike world line are timelike separated, $d s^2<0$ To measure the distance along a particle's world line, it is convenient to introduce
$d \tau^2 \equiv-d s^2 / c^2 .$"

Preceding this paragraph, the author defined that $(ds)^2$ as the squared distance between points in spacetime.
$d s^2=-(c d t)^2+d x^2+d y^2+d z^2$.

I've two questions :

* Why do we say clocks measure timelike distances? Why don't we say clocks measure timelike time? And why insist on 'Timelike'? What about clocks measuring spacelike distances/times?

* If we say that $(ds)^2$ is the squared distance between points in spacetime then the distance is imaginary for timelike events?

 

[PeterDonis]

My book

What book?

 

[PeterDonis]

Why do we say clocks measure timelike distances? Why don't we say clocks measure timelike time?

Both usages are found in the literature. Your book appears to prefer "timelike distance" over "proper time" as terminology. It's the same physics either way.

What about clocks measuring spacelike distances/times?

Spacelike intervals can't be measured by clocks. As far as I know nobody uses "spacelike time" and "timelike time" as terminology instead of "timelike distance" and "spacelike distance". But again, all this is just terminology. The physics is the same either way. Clocks measure timelike intervals; rulers measure spacelike intervals. These are physically different measurements and that difference is reflected in the math as the difference between timelike and spacelike.

* If we say that $(ds)^2$ is the squared distance between points in spacetime then the distance is imaginary for timelike events?

If you insist on mindlessly taking the square root, yes. But nobody does that in practice.

 

[Vanadium 50]

I would prefer the language "timelike interval".

And what book?

 

[Dale]

* Why do we say clocks measure timelike distances? Why don't we say clocks measure timelike time? And why insist on 'Timelike'? What about clocks measuring spacelike distances/times?

Strictly speaking it should be "clocks measure timelike spacetime intervals", but I assume that the book made clear that in that section it was using the word "distances" to refer to "spacetime intervals"

If we say that (ds)2 is the squared distance between points in spacetime then the distance is imaginary for timelike events?

You could say that, but there is no benefit to doing that in terms of the physics. We frequently use imaginary numbers, e.g. in the Fourier transform or in the wavefunction, when there is something physical that corresponds to the imaginary number. But here there is not so we don't usually do it.

 

[robphy]

Since we're talking terminology,
I'd reserve "interval" to be akin to "magnitude of a [path-independent] displacement".

So, an inertial clock between two events A and B
will measure a "proper time along that worldline between A and B"
that is equal to the "[path-independent] spacetime-interval from A to B".
Non-inertial clocks from A to B would measure a shorter proper time along their worldlines.

One could say that clocks do measure tiny infinitesimal-intervals along their worldlines,
but necessarily "the interval" (in my proposed definition) between the endpoints of a worldline segment.

With more detail...
The motivation is that some books use
$${(\rm infinitesimal)}\quad ds^2=dt^2-dx^2$$
or
$${\rm (finite)} \qquad \Delta s^2=\Delta t^2-\Delta x^2=(t_2-t_1)^2-(x_2-x_1)^2.$$
Clocks measure $\int ds$ along timelike-worldlines, which aren't necessarily equal to $\sqrt{\Delta s^2}$.

 

[Dale]

I'd reserve "interval" to be akin to "magnitude of a [path-independent] displacement".

Hmm, I have never liked path-independent quantities in this context. They don’t generalize well.

 

[robphy]

Hmm, I have never liked path-independent quantities in this context. They don’t generalize well.

What would you call $\Delta s^2=\Delta t^2-\Delta x^2$, as if you were defining it in a textbook?

 

[Dale]

What would you call $\Delta s^2=\Delta t^2-\Delta x^2$, as if you were defining it in a textbook?

I would call that the spacetime interval but I would not define it as path independent. I would define it as being the integral of $-dt^2+dx^2$ specifically along the straight-line path and say that I am just being lazy and not explicitly writing the path or the integral that is implied with this notation.

 

[robphy]

I would call that the spacetime interval but I would not define it as path independent. I would define it as being the integral of $-dt^2+dx^2$ specifically along the straight-line path and say that I am just being lazy and not explicitly writing the path or the integral that is implied with this notation.

Ok... the main point of the proposed definition of the
squared-interval is "squared-magnitude of the displacement 4-vector", a scalar.
$\Delta s^2= \Delta \tilde s \cdot \Delta \tilde s$.

(In intro physics, I emphasize that
a distinction between distance and displacement between two points:
the distance depends on the path, the displacement does not.

I'm proposing an analogue for spacetime:
between two timelike-related events,
the elapsed proper-time depends on the worldline, the interval does not.)

 

[vanhees71]

My book writes
" The geometric distinction between timelike and spacelike distances is mirrored in the devices used to measure them. A clock is a device that measures timelike distances; a ruler is a device for measuring spacelike ones. Two nearby points or a timelike world line are timelike separated, $d s^2<0$ To measure the distance along a particle's world line, it is convenient to introduce
$d \tau^2 \equiv-d s^2 / c^2 .$"

Preceding this paragraph, the author defined that $(ds)^2$ as the squared distance between points in spacetime.
$d s^2=-(c d t)^2+d x^2+d y^2+d z^2$.

I've two questions :

* Why do we say clocks measure timelike distances? Why don't we say clocks measure timelike time? And why insist on 'Timelike'? What about clocks measuring spacelike distances/times?

* If we say that $(ds)^2$ is the squared distance between points in spacetime then the distance is imaginary for timelike events?

It's, because the afficionados of a "geometry-only interpretation" of relativistic spacetime models overdo it with the analogies between Euclidean and Minkowskian notions of geometry, although the differences between the two are crucial for the understanding that Minkowskian geometry makes physical sense as a spacetime model, while Euclidean ones can't.

The reason is very fundamental: Only with the signature (-+++) (which convention your book obviously follows) or (+---) for the fundamental form of the spacetime manifold there is (at least locally) a notion of causality, and only time-like separated events can be in causal connection with each other.

"Proper time" is thus defined only for time-like world lines, i.e., such world lines, $x^{\mu}(\lambda)$, (where $\lambda$ is an arbitrary parameter describing the world line) for which the tangent vector is time-like, $g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}<0$ everywhere:
$$\mathrm{d} \tau = \frac{1}{c} \sqrt{-g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}} \mathrm{d} \lambda.$$
Such a world line describes the trajectory of a massive point particle, and proper time is the time an ideal clock measures, which is comoving with this particle.

So you should not take "distance" literally here since indeed, what clocks measure are times along time-like curves. By definition, of course, proper time is a real quantity and not imaginary.

 

[Kashmir]

What book?

Hartle, Gravity

I would prefer the language "timelike interval".

And what book?