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Sorry if this is a dumb question.

A friend of mine just asserted that it's possible to get anywhere in the universe in less than 30 seconds of your time due to time dilation. I do imagine that'll need an incredible amount of energy (perhaps more than is available in the universe if you wanted to cross our galaxy in that time?). Is there a way to quickly calculate how much energy will be needed given the time interval you want to spend traveling to cross a given distance (taking time dilation and everything into account)? Assume my weight is 70Kg.

EDIT: @Ben Crowell's answer is a very good estimate (+1). However, it seems to assume a constant velocity required to cross the galaxy. He starts with the equation ($L$ is the size of the galaxy)-

$$L=v t$$

However, practically we would expect the traveler to start from zero velocity and accelerate all the way to the destination. In this case, the accelerate required will be given by:

$$a = \frac{2L}{t^2}$$

I can't seem to make progress beyond this since I don't know how the $\gamma$ term relates to $a$.

Also, if like @Alexander mentioned, we wanted to decelerate halfway through our journey so we don't destroy our destination, is it fair to say the energy requirement exactly doubles?

Qmechanic
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Rohit Pandey
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  • Can we also assume that you can survive arbitrarily high acceleration and ignore any matter that can stand on your way? Even then, distant regions of universe are already lost to us due to expansion. – Alexander Jun 20 '18 at 23:23
  • Yes, let's assume I'm made of a fictional material that can withstand arbitrarily high acceleration and ignore matter in the way. And let's focus on crossing the galaxy (which I thing is bound by gravity tightly enough that the ends of it are not expanding away from each other). – Rohit Pandey Jun 20 '18 at 23:26
  • Also, would you like to decelerate at the destination, or it's Ok to hit the target going at 0.999999 c? – Alexander Jun 20 '18 at 23:30
  • Good question. How about both cases? Case 1 I'm an astronaut and case 2 I'm a missile :) – Rohit Pandey Jun 20 '18 at 23:32
  • If one of the cases is significantly easier than the other, I'd be happy to know the answer to just the one as well. – Rohit Pandey Jun 20 '18 at 23:34
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    There's useful information and formulae for constant acceleration on the old Usenet Relativistic Rocket page. Wikipedia has similar info in an article with the same name. And if you want to know how to derive the formula for relativistic speed with constant acceleration, see the end of my answer. – PM 2Ring Jun 21 '18 at 02:58
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    As a table in my 1st link shows, the amount of energy required is much greater if you want to deceleration for half the trip so you arrive at rest at the destination. – PM 2Ring Jun 21 '18 at 03:01

1 Answers1

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This is a cute question, +1. I feel the urge to make it into a homework problem for my poor, unsuspecting students.

Let the galaxy have size $L$, let $\tau$ be the proper time required to cross the galaxy at constant velocity, let $t$ be the time required in the galaxy's rest frame, let $K$ be your kinetic energy, and let $m$ be the mass of you and your spaceship. In natural units, where $c=1$, we have

$$v=L/t$$

$$K=m(\gamma-1)$$

$$\tau=t/\gamma.$$

The solution of these equations is

$$K=m\left[\sqrt{1+\left(\frac{L}{\tau}\right)^2}-1\right],$$

or, reinserting factors of $c$,

$$K=mc^2\left[\sqrt{1+\left(\frac{L}{c\tau}\right)^2}-1\right].$$

For $m=70$ kg and $\tau=30$ s, the result is $\sim10^{30}$ J, or something like $10^{14}$ megatons of TNT. Your ultrarelativistic friend's body has so much kinetic energy that if he collided with the earth, it would be the end of the world, so I think Congress should pass a law prohibiting him from doing this.

  • One way to avoid the end of the world would be decelerating the mass. However, this would double the total amount of energy. – Alexander Jun 21 '18 at 00:31
  • Also, in this answer you work with a constant velocity, $v$. However, practically we would either accelerate to $v$ and then decelerate or accelerate all the way to the destination. Is it possible to take the acceleration into account? If not, is it safe to say that taking it into account (for both cases) would increase your estimate? – Rohit Pandey Jun 21 '18 at 00:37
  • In classical physics, $t = \sqrt{\frac{2L}{a}}$ – Rohit Pandey Jun 21 '18 at 00:42
  • BTW, if I ever learn physics from you, I'll be sure to comb through all your physics SE activity :) – Rohit Pandey Jun 21 '18 at 00:48
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    By comparison, according to Wikipedia, the total primary energy production of the world for 2013 was $5.67×10^{20}$ joules. So that quick trip across the galaxy costs 2 billion years worth of global energy. – PM 2Ring Jun 21 '18 at 03:11
  • @PM2Ring - Maybe I'm wrong, but I suspect this estimate is a lower bound (see edit to my question). – Rohit Pandey Jun 21 '18 at 06:03