Rindler - uniform acceleration

[Lapidus]

So a light signal is sent off some space behind me. At the same time I start accelerating extremely quickly. Even though the light signal will always be faster than me it will never catch up with me.

I have difficulties to understand that something that is always faster than you can still never catch up with you. I know of course the Rindler diagram how the curve of the accelerating spaceship gets closer and closer to the light signal. But is there another way to understand this than by looking at a diagram? How can something that is forever faster than you never catch up with you?

thank you

 

[PeterDonis]

How can something that is forever faster than you never catch up with you?

Because it started off far enough behind you that its speed advantage--which continually decreases as you accelerate--is never enough to make up the distance.

 

[pervect]

So a light signal is sent off some space behind me. At the same time I start accelerating extremely quickly. Even though the light signal will always be faster than me it will never catch up with me.

I have difficulties to understand that something that is always faster than you can still never catch up with you. I know of course the Rindler diagram how the curve of the accelerating spaceship gets closer and closer to the light signal. But is there another way to understand this than by looking at a diagram? How can something that is forever faster than you never catch up with you?

thank you

Just look at the behavior of a hyperbola.

Consider two curves: Curve C1 which has satisfies the equation $s = t$, where s is distance and t is time. This is a straight line through the origin, with a slope of one.

Next consider the curve C2 which satisfied the equation $s = \sqrt{t^2 + 1}$, (you could also say it satisfied the equation $s^2-t^2 -t^2=1$). Curve C2 is a hyperbola. Using calculus, we can calculate the slope of this curve, which represents the speed of an object following this curve.

$$\frac{ds}{dt} = \frac{t}{\sqrt{1+t^2}}$$

We then make two observations.

An object following curve c1 never "catches up" with an object following curve c2. For any time t, C2 > C1, as $\sqrt{1+t^2} > t$. If this doesn't seem obvious, square both sides.

The slope of the curve c2 is always less than 1, while the slope of curve c1 is always equal to 1. Thus the slope of curve c2 is less than the slope of curve c1, or the speed of an object following c2 is always less than the speed of an object following curve c1. This again follows from the fact that $\sqrt{1+t^2} > t$

We have previously noted that an object following curve C1 can never catch up with an object following curve C2, even though we've just proved that the object following C2 is always slower than the object following curve c1.

If we draw out curve C2, we'd say that it's a hyperbola. And curve C1 is the asymptote of the hyperbola. And we'd say that the hyperboa approaches the asymptote ever more closely as time increases, but never reaches it.

Uniformly accelerated motion is called hyperbolic motion, and in fact curve C2, with the right choice of origin and scale, is a possible motion for a uniformly accelerating observer.

 

[Umaxo]

I would add one more explanation from little different point of view:

Look at it from the point of your initial inertial frame of reference where you started to accelerate. In this frame, you will get relativly quickly to 0,99% of speed of light, then 0,999% speed of light and so one. So the difference between your speed and the speed of light is getting smaller and smaller (in this particular frame of reference). Thus you get to something very similar to Zenos paradox of Achilles and the tortoise.