Newie relativity question: proper time vs. arc length?

[Peeter]

Doran/Lasenby define a proper interval as:

$$\delta \tau = \int \sqrt{\frac{dx}{d\lambda} \cdot \frac{dx}{d\lambda}} d\lambda$$

(c=1, x= (t,x1,x2,x3) is a spacetime event, and the dot product has a +,-,-,- signature)

and say that this is called the proper time.

I can see that this would be preferred as a parameterization in the same way that arc length is a natural parameterization of a curve (takes out zeros in the derivatives if one ``stalls'' for a while along the curve with respect to some specific parameterization).

However, calling this "time" seems slightly perverse to me since it looks to me (from an math point of view) as not much more than arc length with respect to this particular dot product.

Except for a particle is at rest, I can't imagine why one would want to call this parameterization time (unless its just because spacetime-arc-length is a mouthful). Can anybody else account for why this may be a reasonable choice of nomenclature?

 

[Dale]

If a clock moves along some arbitrary (timelike) path in spacetime then the elapsed time will be equal to this arc-length. In other words, another way to look at the twin paradox is that the traveling twin shows less time on his clock simply because his spacetime-arc-length is shorter than the home twin.

 

[lbrits]

Yes it is perverse but as DaleSpam pointed out, it is in fact a true statement and a foundation of relativity. You can build your path out of many short, straight segments, and use the idea of Lorentz transformations to convince yourself that the voyager's time ellapsed along each segment is given by $$\frac{1}{\gamma} dt = \frac{d\tau}{dt} dt = d\tau$$, so essentially integrating along that curve is equivalent to summing up all those little straight segments.

 

[robphy]

Proper-time is about 100 years old this year... due to the mathematician Minkowski.

Many results in Special Relativity can be interpreted geometrically, with a different metric (aka dot product)...with analogous geometrical constructions. The underlying reason for this is that the geometry of Euclidean space and that of Minkowski space are examples of Cayley-Klein geometries.

 

[Peeter]

Thanks for the explanations guys. I guess it's not perverse if the elapsed time along that tragectory equals the arc length. I'll have to do some calculations with this to get the feeling for how it fits together.

Now, the twin paradox wasn't something I planned to think about anytime in the near future, but perhaps it's unavoidable. My primary goal for looking at relativity is to understand how it's related to Electromagntism so integrally. E&M is a subject I can relate to (I can watch TV, look with my eyes, use a radio, cell phone, ...), but the twin paradox isn't something that I can compare to something that I actually observe or know.

Is there a good example short of getting a really fast spaceship that physically demonstrates clocks going out of sync after taking different spacetime paths from a fixed point to some other later time common fixed point in space. Perhaps some sort of particle collider observable or an E&M phenomena that I "know about" but don't realize how it actually demonstrates this?

 

[robphy]

Is there a good example short of getting a really fast spaceship that physically demonstrates clocks going out of sync after taking different spacetime paths from a fixed point to some other later time common fixed point in space.

Consider the light-clock.
Here is a fancy visualization I made:
http://physics.syr.edu/courses/modules/LIGHTCONE/LightClock/

 

[Dale]

Is there a good example short of getting a really fast spaceship that physically demonstrates clocks going out of sync after taking different spacetime paths from a fixed point to some other later time common fixed point in space. Perhaps some sort of particle collider observable or an E&M phenomena that I "know about" but don't realize how it actually demonstrates this?

Yes. Look at the http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Twin_paradox"section in the FAQ. Muons in a storage ring at relativistic speeds have a longer half-life than muons at rest in the lab frame, as measured in the lab frame. Their spacetime-arc-length is shorter than their resting "twin's".

 

[Peeter]

You can build your path out of many short, straight segments, and use the idea of Lorentz transformations to convince yourself that the voyager's time ellapsed along each segment is given by $$\frac{1}{\gamma} dt = \frac{d\tau}{dt} dt = d\tau$$, so essentially integrating along that curve is equivalent to summing up all those little straight segments.

I tried a simple version of this calculation using just the dot product, using a linear velocity scenerio (I've put the c's back in ... more confusing with c=1 for me at least now):

$$x = x^0 \gamma_0 + x^1 \gamma_1$$

where:

$$x^1(t) = vt + x^1(0)$$

$$x^0(t) = ct$$

So,

$$\frac{dx}{dt} = (c,v)$$

So,
$$\frac{dx}{dt} \cdot \frac{dx}{dt} = c^2 - v^2$$

Thus

$$d\tau = c \sqrt{1 - (v/c)^2} dt$$

Or,

$$d\tau = c \frac{1}{\gamma} dt$$

So, sure enough this arc length is a measure of elapsed time, but one that is scaled by the velocity involved.

It appears that it's normal in relativity texts to think of this the other way around... ie: the traveller's time is scaled locally relative to this arc length (``proper time'').

 

[Peeter]

It appears that it's normal in relativity texts to think of this the other way around... ie: the traveller's time is scaled locally relative to this arc length (``proper time'').

I think I get it. It's not just a matter of a different way to think about it. The experiments (muon decay rates, atomic clocks on planes, ...) where the rate of events when you have motion involved are scaled this way show that the event arc length is the more fundamental concept, and the local concepts of time (or length) measured separately are only approximate.