# Why can't we travel at the speed of light? [closed]

I understand we may not have the technology for it as of now. But I have also heard it's because we don't have enough energy, and traveling at or near the distance of light speed will require an almost infinite amount. My question is this the only thing stop us from traveling at the speed of light?

## closed as off-topic by Everyone, Hohmannfan♦, Brian Tompsett - 汤莱恩, ForgeMonkey, kim holderApr 13 '16 at 13:54

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• "This question is about other space sciences (physics, weather, astronomy, etc), and does not directly pertain to space exploration as outlined in the help center." – Everyone, Hohmannfan, Brian Tompsett - 汤莱恩, ForgeMonkey, kim holder
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• What fraction of the speed of light do you have in mind? – Rikki-Tikki-Tavi Apr 12 '16 at 21:03
• "My question is this the only thing stop us from traveling at the speed of light?" Isn't that enough? – Organic Marble Apr 12 '16 at 22:00
• It turns out that a couple of quick google searches on this didn't show up anything useful near the top, but then typing in a question like the one you asked did much better. Why you can't travel at the speed of light seems good, also Frequently Asked Questions About Special Relativity. – kim holder Apr 12 '16 at 22:18
• "..at the speed of light? ..will require an almost infinite amount." No, travelling at the speed of light would require an infinite amount of energy according to Relativity Theory! – Andrew Thompson Apr 12 '16 at 23:46

We simply have to look at the formula relativistic kinetic energy to see why we can not travel at the speed of light:

$$E_k = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}-mc^2$$

Where $m$ is the rest mass, $v$ is the velocity, and $c$ the speed of light. If we try to set $v=c$, we get the following:

$$E_k = \frac{mc^2}{0}-mc^2$$

We can not divide by zero. Nope.
It is not just an "almost infinite amount" of energy, it is infinite.

(Note: Things like photons that travels at the speed of light elegantly avoids this problem by not having any mass at all.)

• I'd argue "elegantly" since they get a $0 \over 0$ into these equations and you need to reach completely elsewhere (quantum physics) to derive their (non-zero and finite) energy. But the concept is there: spend a fixed amount energy to halve the remaining gap between your speed and the speed of light. Spend second that much to halve what remains. And so on, you never get there. – SF. Apr 13 '16 at 7:09

Hohmannfan neatly answered the "AT" speed of light part. Let me bite the "Near" one.

Let's pick a target speed of 225,000 kilometers per second, or 0.75c.

It's a kinda neat number which I like to use because had Newton been right and Einstein wrong, the same kinetic energy of a body moving currently at 0.75c, would give it just a notch above 1c in Newtonian mechanics.

Let's build an ideal rocket engine, that uses matter-antimatter annihilation (the absolutely most energetic reaction currently known to man, and unlikely to be topped by any other, ever.) and ejects the reaction mass (or in this particular case reaction energy at speed of light.

The Specific Impulse would be c.

(all units are off by the $g0$ factor, a constant of 9.8m/s^2 of Earth's gravitational acceleration; it doesn't really matter here as we obtain a dimensionless factor in the end anyway)

The same article gives relativistic $\Delta v = c \cdot \tanh \left(\frac {I_{sp}}{c} \ln \frac{m_0}{m_1} \right)$

Let's get the mass fraction, $M = \frac{m_0}{m_1}$.

$M = e^{{c \over I_{sp} }{\tanh^{-1}( {\Delta v \over c} )}}$

${c/I_{sp}}$ is conveniently 1, and $0.75c \over c$ is 0.75. So, $M = e^{\tanh^{-1}(0.75)}$

Our mass fraction comes down to about $m_0 = 2.65 m_1$.

That means for 1 ton of rocket, you need 1.65 ton of propellant.

It's very good given rocket standards. For chemical rockets this comes up as something of order of 95-98 tons of propellant for a ton of cargo.

And still, that means a rocket of 1 ton (a very conservative number) which contains everything, including generators of magnetic field to contain the antimatter, and the annihilation-engine, on top of whatever else it needs (guidance? RCS? Any actual payload?) needs 825kg of propellant matter, and another 825kg of propellant antimatter.

Current estimate is \$1bln per 40 milligrams of positrons*. So, for the needed 825kg... \$20,625,000,000,000,000 or 20 quadrillion dollars? The world economy is about two orders of magnitude short; if you sold everything in the world that can be sold, you'd still come up way short.

Never mind the engine will NOT be 100% efficient. Never mind we have absolutely no clue how to store such amounts of antimatter, especially in anything that weights less than the stored antimatter (or even less than a million times more than the stored antimatter...).

And that's the ONLY engine that can get us near relativistic speeds.

Let's take the next big candidate: Nuclear fusion.

Same source as before, ISp = 0.119c. Again, but ${c \over I_{sp}}$ is no longer 1. It is now 8.4.

$M = e^{{c \over I_{sp} }{\tanh^{-1}( {\Delta v \over c} )}} = e^{{c \over 0.119c }{\tanh^{-1}( 0.75 )}} = e^{8.4 \cdot 0.973} = e^{8.17} = 3533$

And so, with 8.17 instead of 0.97 we had before, our neat exponent blows up to over 3500. We are below a 0.03% dry mass fraction.

That means for every ton of equipment, fusion reactor, engine that exploits the products of fusion, even wall of a tank - you need to pack 3500 tons of propellant. Can you even store 3500 tons of hydrogen in a tank that weighs less than a ton?

That's not something that scientists and engineers can improve upon. That's the end of the project, a simple, flat "impossible".

This is what is known as "a victim of the tyranny of the rocket equation", a design that is completely and irrevocably impossible from practical point of view, because we don't foresee any storage technology of hydrogen that would give us a 1:3500 weight ratio between the tank and the stored hydrogen, never mind giving any wiggle room for the actual payload, engines, reactor.

It would be easier to design an annihilation-based rocket from scratch, than one that would keep this mass ratio viable - never mind developing nuclear fusion technology that squeezes every last possible joule off the fusion, without losses. And the losses would likely drive our payload fraction another order of magnitude down.

• let me add one more perk of that chosen 0.75c: the time and space dilation at that speed plays in such a way, that travel time, from the point of view of the crew of the spaceship is as if they were traveling just over 1c. While visuals, distances, etc get all skewed, if you travel from Earth to Proxima Centauri, 4 light years away, and you travel at 0.75c (on the average) you'll be older by 4 years when you get there. (also, if you ramp up your speed until halfway, then decelerate, both at such rate that 0.75c is your average speed, you'll have about 1g of acceleration...) – SF. Apr 19 '16 at 10:03