Lorenz transformation question (1 Viewer)

sida1049 · Dec 5, 2015

Since evolution has conditioned the human body (and almost all other land animals) to only be able to withstand 1g for a long period of time, here's my question:

How long would it take for a person to accelerate to near the speed of light (for convenience, assume that we are accelerating the subject to the speed of light) at a constant acceleration of 9.8m/s^2 according to the frame of reference of the person? Assume that the speed of light is 300,000,000m/s.

This is merely out of curiosity. All necessary working out would be appreciated.

InteGrand · Dec 5, 2015

sida1049 said:
Since evolution has conditioned the human body (and almost all other land animals) to only be able to withstand 1g for a long period of time, here's my question:

How long would it take for a person to accelerate to near the speed of light (for convenience, assume that we are accelerating the subject to the speed of light) at a constant acceleration of 9.8m/s^2 according to the frame of reference of the person? Assume that the speed of light is 300,000,000m/s.

This is merely out of curiosity. All necessary working out would be appreciated.

See the answers here:
http://www.thenakedscientists.com/HTML/questions/question/1000139/

InteGrand · Dec 5, 2015

$Also check out John Baez's excellent article on this matter here:$

http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

$He discusses what we mean by a ``uniform acceleration'' and provides the useful equations, but doesn't derive them there. For a derivation, see Section 6.2 in \emph{Gravitation} by Misner, Wheeler and Thorne:$

http://www.amazon.co.uk/Gravitation-Physics-Charles-W-Misner/dp/0716703440

$Alternatively, try searching for the derivations online.$

$The relevant equation for you for your question is this one:$

$v = c\tanh \left(\frac{aT}{c}\right) = \frac{at}{\sqrt{1 +\left(\frac{at}{c}\right)^2}}$

$where the meanings of the symbols are explained in the link ($a$ is the acceleration, $T$ is the time in the rocket's frame of reference, and $t$ is the time in the observer's frame of reference, e.g. on Earth. Obviously $c$ is the speed of light.). Also, $\tanh$ is the hyperbolic tangent function: $\tanh x =\frac{\mathrm{e}^x -\mathrm{e}^{-x}}{\mathrm{e}^x +\mathrm{e}^{-x}}$, where the domain of the function is $\mathbb{R}$ and the range is $(-1,1)$ (you can search up a graph of it to aid visualisation).$

$We can use this equation with $a =1g$ to play around to see what time $T$ the astronauts would age to reach given fractions of light speed. Use $T= \frac{c}{a} \tanh ^{-1} \left(\frac{v}{c}\right)$, where $\frac{v}{c}\in (0,1)$ is the fraction of light speed you want and $\tanh^{-1} x =\frac{1}{2} \ln \left( \frac{1+x}{1-x}\right)$ for $x\in (-1,1)$ is the inverse hyperbolic tangent function.$

$In fact, Baez has already played around for us and made a table with required times in years (and corresponding distances in light years) to reach certain speeds at $a=1g$. He also addresses many other questions like fuel requirements etc., so well worth a read of that article.$

sida1049 · Dec 5, 2015

Thank you!

InteGrand · Dec 5, 2015

sida1049 said:
Thank you!

$You're welcome! Also, Wikipedia has the derivation of the equation for $v$ given on John Baez's page. In fact, Wikipedia derives the equation for $\Delta v$, but this will just be equal to our final speed $v$ if we start with an initial rocket speed of $0$, which is what we'd expect to do.$

https://en.wikipedia.org/wiki/Relativistic_rocket#Formula_for_.CE.94v .

$That derivation makes use of the classical rocket equation derived by Tsiolkovsky:$

https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation .

$It also makes use of the Lorentz transformation, so you may want to read up on that if you haven't already$

https://en.wikipedia.org/wiki/Lorentz_transformation .

$The final equation in the Wikipedia derivation in the first link has the form $\Delta v = c \tanh \left(\frac{I_{sp}}{c}\ln \left(\frac{m_0}{m_1} \right) \right)$, where the meanings of the symbols are explained in the link (basically, this relates the change in speed in a relativistic setting with the change in mass of the craft (obviously the mass decreases due to consumption of fuel)). However, it was derived in an earlier part of the derivation that $\frac{m_0}{m_1}=\exp \left(\frac{aT}{I_{sp}} \right)\Longleftrightarrow \ln \left(\frac{m_0}{m_1}\right) = \frac{aT}{I_{sp}}$, where $T$ is the proper time, $a$ is the constant acceleration and all symbols are defined in the derivation. Substituting this in to the $\Delta v$ equation and cancelling $I_{sp}$'s results in the equation we had from Baez's article.$

$(Note that the Wikipedia derivation uses $t$ for the proper time, whereas Baez used $T$. Wikipedia uses $t^\prime$ for the rest frame time, whereas Baez uses $t$ (which is what Wikipedia used for \textsl{proper} time), so keep this in mind as these disparities can cause confusion.)$

Lorenz transformation question (1 Viewer)

sida1049

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InteGrand

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