Constant acceleration in a rocket

AI Thread Summary
The discussion focuses on analyzing constant acceleration in a rocket using tensor formalism and Lorentz transformations. The author derives the four-vector acceleration in the rocket's frame and translates it to the Earth's frame, ultimately expressing velocity and position as functions of time. It is noted that as time approaches infinity, the rocket's speed approaches the speed of light, but it does not imply that the rocket actually reaches this speed. The conversation also touches on the implications of this acceleration on communication between the Earth and the rocket, emphasizing that after a certain time, signals from Earth cannot reach the rocket. The analysis concludes with a graphical representation of the rocket's trajectory, illustrating that it never intersects with the path of light signals.
Frostman
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Homework Statement
A rocket, starting from the earth, moves away from it subjected to ##a## constant acceleration in the reference system of instantaneous rest of the rocket itself.
a) Obtain the law of motion, that is the dependence of the distance of the rocket from the earth, as a function of the earth's time ##t##.
b) Show that there is a maximum time ##T## after departure, after which it is impossible to send messages capable of reaching the rocket from the ground. Calculate the dependence of ##T## on ##a##
Relevant Equations
Tensor formalism
I thought I'd start by writing the problem in a tensor formalism. I have identified with ##S## the Earth and ##S'## the rocket. Since the acceleration provided is in the rocket's frame of reference, I can write the following four-vector.
$$
a'^\mu=(0, a, 0, 0)
$$
Since we are interested in the equation of motion in the frame of the earth, I go directly to the frame of reference ##S##. Applying the Lorentz transformations.
$$a^0 = \gamma(a'^0+va'^1)=\gamma v a$$$$a^1 = \gamma(a'^1+va'^0)=\gamma a$$
But I get the four-vector acceleration starting from the derivative of the four-vector velocity with respect to proper time.
$$a^\mu = \frac{d}{d\tau}(\gamma, \gamma v_x, \gamma v_y, \gamma v_z)=(\gamma \frac{d(\gamma)}{dt},\gamma \frac{d(\gamma v_x)}{dt},\gamma \frac{d(\gamma v_y)}{dt},\gamma \frac{d(\gamma v_z)}{dt})$$
In particular
$$a^0 =\gamma v a=\gamma \frac{d(\gamma)}{dt}$$$$a^1=\gamma a = \gamma \frac{d(\gamma v_x)}{dt}$$
Considering the second result:
$$\int_0^v d(\gamma v) = a\int_0^t dt$$
So we have:
$$ \frac{v}{\sqrt{1-v^2}}=at$$$$v(t)=\frac{at}{\sqrt{1+a^2t^2}}$$
By integrating speed over time we have:
$$x(t)=\frac 1a \bigg(\sqrt{1+a^2t^2}-1\bigg)$$
Can it be exhaustive?
The interesting thing is that for big times (##t\rightarrow +\infty##), the speed tends to ##1##. So it comes back to me as a result.
For the second point how could I operate? From this time ##T##, since no luminous message sent from Earth will never be able to reach the rocket, it means that the rocket is also traveling at speed ##1##. I would conclude that this time ##T## is for ##T\rightarrow +\infty##. But it seems trivial and stupid (because I have no dependence from ##a##), in what terms should I reason?
 
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Frostman said:
For the second point how could I operate? From this time ##T##, since no luminous message sent from Earth will never be able to reach the rocket, it means that the rocket is also traveling at speed ##1##.
It doesn't imply that the rocket is ever traveling at the speed of light.
 
PeroK said:
It doesn't imply that the rocket is ever traveling at the speed of light.
Okay, taking a step back, there is a time, after that time the messages sent from the Earth will never reach the rocket.
Can I see it in terms of equations of motion? Since the position occupied by the rocket will be after that time T, always greater than the position occupied by the signal.
 
Try plotting x(t) on a spacetime diagram.
 
Okay, it's an hyperbola.
Furthermore, the more the acceleration value is greater, the more the trend is that of: ##x(t)=|t|## and this one is the trend of the light signal.
The two trends, as I have drawn them, never meet.
 
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