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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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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.