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Useful Notes: Relativity
This article needs Wiki Magic.

Albert Einstein's Theory of General Relativity is perhaps the second-most confusing subject in all of modern physics, beaten only by Quantum Mechanics. E = MC Hammer applies to this in full force, meaning very few people understand all the incredibly weird results that Einstein's very, very simple ideas imply. (Writers Cannot Do Math means that most of the media isn't helping.)

Relativity was worked out by Albert Einstein in two pieces, special relativity and general relativity. Bright high schoolers can handle the math of special relativity. The math of general relativity makes graduate students cry, so we'll leave it to the other wiki.

The speed of light

Starting with the ancient Greeks, and ending around the seventeenth century, scientists and philosophers argued extensively about whether light had a speed, or whether it moved infinitely fast. Galileo tried to measure the speed of light with lanterns and mirrors on mountaintops, but failed. In 1676, a Danish astronomer named Ole RÝmer discovered, by observing a solar flare and timing how long it took for the brighter light to be reflected off Jupiter's moons, that light did indeed have a finite speed. Later scientists created successively more and more precise means of measuring the speed of light, and today we can measure its speed with such great precision that the meter is defined in terms of the speed of light rather than the other way around. The speed of light in a vacuum is exactly 299,792.458 kilometers per second.note 

That means that you can move nearly as fast as light. And when you do, things start to look strange. (Check out this movie to see just how strange.) (Or even this prototype game to experience it first hand.)

These effects are only visible if you move near the speed of light, so they're often viewed as being part of relativity. However, a lot of them are really due just to the fact that light has a finite speed. A bat flying at near the speed of sound would notice the same sort of effects, and sound travels far too slowly for relativity to have an effect.

     What does it look like to travel near the speed of light? 

Let's suppose that the ''Enterprise'' flies straight from Earth to Mars, speeding up the whole way.note  Guinan looks out a window at the front. Chief O'Brien looks out of a window at the back. Jack O'Neill brings his telescope along and looks out the side window. What do they see?

Their view will be drastically different than the view from a slower ship.

First, the Doppler shift will change the colors of everything involved. Everything ahead of the ship will be blue-shifted, and everything behind the ship will be red-shifted.

As seen from Earth, Mars is red. As the ship speeds up, the Mars Guinan sees will change color: it will first appear orange, then yellow, then green, then blue, then violet, then (if Picard decides to go very very fast) Mars will disappear entirely into the ultraviolet. Meanwhile, the same thing is happening in reverse to O'Brien: Earth will go from blue to green to yellow and on up.

If the ship accelerates while O'Brien is close enough to Earth to see the ocean and the forest, then the forest will change color "ahead" of the ocean. That is, if Earth is redshifted enough that the oceans look yellow, the forests will all look orange.

(The terms "blue-shift" and "red-shift" are a bit misleading. "Blue-shifted" means "moved towards blue", not "becomes blue-tinged". Science fiction movies sometimes get this wrong. Perhaps the most nonintuitive example is what happens to the color violet. Due to oddities about the human eye, violet light appears as a mix of red light and blue light, despite the fact that violet lies on the extreme "blue" end of the spectrum. Thus, when violet light is "red-shifted", it becomes blue—and thus, to humans, looks less red. In addition we should note that the spectrum of stellar light is quite broad, and so a star might look redder if the blue part of its emissions shifts into invisible ultraviolet, or if previously invisible infrared gets shifted into visible red light.)

Second, Mars will get brighter and Earth will get dimmer. Here's one way to think about this. Suppose Boothby stays behind on Earth and flashes one pulse of light at the Enterprise every second. Since the Enterprise is moving away, a second after O'Brien sees the first pulse, the ship will have moved away from Earth, so the second pulse won't have caught up—it will take more than a second for him to see the second pulse. If you picture now a strobe light flashing bursts every millisecond, then from O'Brien's viewpoint, this delay will "smear out" until it looks like a steady, dimmer light.note 

You can hear these effects with sound waves. Try listening to an ambulance: it will sound higher-pitched and louder as it comes towards you, and lower-pitched and softer as it runs away, for exactly the same reasons.

Jack's view out the sides of the ship is even more bizarre. Let's suppose that he wants to look at Jupiter. And let's suppose that Jupiter happens to be on the right—if the Enterprise stopped, Jack could point his telescope straight out the window and be aimed at Jupiter.

But the Enterprise is moving. Suppose that Jack does point his telescope straight out the window. A photon coming from Jupiter will come in at right angles. If it passes through the front of the telescope, by the time the photon gets to the back of the telescope, the telescope will have moved; it won't get out the back and into Jack's eye. He has to aim his telescope forwards if he wants to see Jupiter.

This happens to everything in the visible universe. Everything, from the Enterprise's perspective, appears to get squished up towards the ship's destination. Mars gets squeezed down to a tiny dot, and Earth gets spread out; everything else in the universe seems to twist towards Mars as the ship speeds up. This is another reason why the view from the front is probably brighter than the view from the back; there's more stuff in it. The term for this effect is the "aberration of light".

(Jack should notice one relativistic effect. Jupiter shouldn't look round. It should be squeezed into a tall, thin oval. From Jack's perspective, Jupiter is Lorentz contracted; he should be seeing it look thinner because it is thinner. However, the aforementioned aberration exactly counteracts this effect: Jupiter still looks perfectly round, only its surface features would be irrevocably distorted.)

Special Relativity

There are two facts that give rise to most of special relativity:

  • There's no such thing as "standing still". If I am moving a thousand miles per second faster than you are, the physics that I see are every bit as good as the physics that you see.
  • Light is a wave that does not need a medium to travel in.

These facts have a lot of interesting and bizarre consequences for what happens when you move near the speed of light. The folders below contain explanations of why these things happen and a history of how they were discovered.

These consequences include:

  • Light always travels at the same speed. Whether you're moving toward the light source, moving away from the light source, or at rest with respect to the light source, you will always measure its light coming toward you at the same speed.
  • Time Dilation: the faster you go, the slower time goes.
  • No simultaneity: Any spaceship that can move faster than light is a time machine.
  • Lorentz contraction: A ship going very fast appears (to outside observers) to be squished in the direction of travel. Conversely, if you move near the speed of light, the distance to your destination actually seems to get shorter.
  • Momentum: The easy way to think about this is that as an object gets nearer the speed of light, it gets heavier.note  So a pebble traveling near the speed of light hits with enough force to cause craters, and accelerating to near the speed of light takes a lot of fuel. (Accelerating to the speed of light would take an infinite amount of fuel.)
  • E=mc2.
    • This above equation only applies if velocity can be assumed to be zero. For a mass at nonzero velocity the appropriate equation becomes E2=m2c4+p2c2
      • And then the reason that it simplifies down to E=mc2 is that the full equation has E as the hypotenuse of a right-triangle, and applies the Pythagorean theorem. E is the hypotenuse, and the legs are mc2 and pc, respectively.

Some of these consequences (length contraction, light traveling at the same speed) do not show up in fiction very often because it's very hard to actually write a plot where they matter.

Some of them (simultaneity and the finite-speed effects above) should show up, but often don't. You can Hand Wave them away, but usually, it's just because the writer skimped on research.

And some of these consequences (Time Dilation and increased mass) do show up, and have much bigger effects than they really should.

If you speed up in a car, you get pushed back in your seat; acceleration, or speeding up, feels a lot like being pulled down by the force of gravity.note  Avoiding this effect is the entire point of Inertial Dampening. Since the force of gravity accelerates you downwards at 10 meters/second/second, if you speed up faster than that, you feel very heavy.

In Planet of the Apes, George Taylor travels away from Earth for 2,000 years. Presumably he spent half his time speeding up and half his time slowing down. If you accelerate at 10 meters/second/second for 1,000 years, you get up to 99.999955% of the speed of light. Time dilation does take effect, so George Taylor should have aged rather less than 2,000 years. However, it takes a while to kick in—the astronauts should have experienced a trip of about 14 years, not 18 months.

     The mathematical formulas for these effects 

Mathematically, the factor by which time slows down, distances in the direction of travel shrink, and momentum increases, is called "gamma" (γ). The formula for gamma is:

γ = 1 / sqrt (1 − v2/c2)

... where "sqrt" means "square root", v is the velocity of travel, and c is the speed of light (300,000 km/sec).

At half the speed of light, γ works out to 1.1547. At 86.60254% of the speed of light, γ is 2. At 99% of the speed of light, γ is 7.0888.

Here's how you use the number γ. If you travel at 99% of the speed of light, then γ is pretty close to 7.

If you travel at 99% of the speed of light for a period of time that feels like 1 year to you, when you get back to Earth, 7 years will have passed there.

If you travel at 99% of the speed of light in a spaceship that's 700 meters long, to everyone on Earth, it will look like your ship is only 100 meters long.

Remember that if a 1-gram pebble traveling at 99 meters per second crashes into a 98-gram rock, the 99-gram accumulation will move at 1 meter per second. This is known as conservation of momentum, and is pretty simple in Newtonian physics.

If a 1-gram pebble traveling at 99% of the speed of light crashes into a 98-gram rock, the 99-gram accumulation will not move off at 1% of the speed of light. It will move off at 7% of the speed of light, because the 1-gram pebble hits harder than you'd expect.

There's a pair of useful formulas that involve γ. They are called the "change-of-frame" formulas.

Suppose that Captain Picard and William Adama fly past each other. They are in deep space, far from any planets, so they have no notion of who is "really" standing still and who is "really" moving. The Enterprise passes the Galactica, so all they know is that the Enterprise is going some amount v faster than the Galactica.

Suppose that the two crews agree that they met at time 0 and at space-coordinates (0,0,0). They agree to call the direction the Enterprise is flying the x direction. The other two space directions are the y direction and the z direction.

In relativity, it doesn't really make sense to talk about the coordinates of places; it only makes sense to talk about the coordinates of events. So both crews agree that the event "the Enterprise passed the Galactica" happened at coordinates (0,0,0,0).

From Adama's point of view (in Adama's "rest frame"), every event that happens on the Galactica happens at coordinates (0,0,0,t), for some number t equal to whatever the Galactica's clock says when the event happens. And from Adama's point of view, every event that happens on the Enterprise happens at coordinates (vt,0,0,t) for some number t.

Similarly, in Picard's rest frame, every event that happens on the Galactica happens at coordinates (-vt,0,0,t), and every event that happens on the Enterprise happens at coordinates (0,0,0,t) for some number t.

Now let's suppose that a supernova goes off. In Adama's rest frame, the supernova goes off at coordinates (x,y,z,t). In Picard's rest frame, the supernova goes off at coordinates (γ(x-vt),y,z,γ(t-vx/c2)).

There is one important thing Adama and Picard agree on. If two events happen at coordinates (X,Y,Z,T) and (X+x,Y+y,Z+z,T+t), the "interval" between the two events is given by the square root of x2+y2+z2-(ct)2.

If the squared interval between two events is positive, then the two events are too far away for light to get from one to the other. If you are present at both events, then you had to travel faster than light.

It turns out that if you make the change of reference frame, the interval between two events may change...but the squared interval will always stay positive or stay negative. That is, if Adama thinks that you had to travel faster than light to get from one event to another, Picard will always agree with him.

Finally, suppose that a spaceship accelerates (in its own reference frame) at g meters/second/second. It starts out at rest relative to some nonaccelerating observers. After t seconds, where t is measured in the reference frame of the nonaccelerating observers, the ship has traveled

x=sqrt(c4/g2 + c2t2) - c2/g

meters, it has a velocity of

v=gt/sqrt(1+g2t2/c2)

meters/second, and the ship has felt

T=(c/g) * ln (gt/c + sqrt(1+g2*t2/c2))

seconds pass. The 14-year travel time mentioned for Planet of the Apes above was obtained by plugging in t=1000 years (to speed up) and g=10 meters/second/second, and doubling (to slow back down so you don't crash into your destination).

     Why light always travels at the same speed 

...and why that is weird.

In classical physics (as opposed to quantum physics), there are two different kinds of things: waves and particles.

An example of a particle is a bullet. If a gun shoots bullets at 1000 miles per hour, and you fire it from a car that's going 100 miles per hour, then the bullet goes 1000 miles per hour relative to the car, and so it goes 1100 miles per hour relative to the ground.

An example of a wave is, well, a wave in water. If you move a stick up and down through the surface of the pond, and watch the ripples spread out, then the ripples always spread out at the same speed, relative to the water. If you move the stick faster, or if you drag it through the water while moving it up and down, the ripples will become closer together or farther apart, but they will always move at the same speed.

Waves travel at the same speed in all directions. So if I'm in a helicopter hovering over a lake, waves going past me north to south should move at the same speed as waves going past me east to west. However, if I'm flying due north, then waves traveling north should seem to go slow relative to me. So I can tell whether I'm holding still relative to the water, by seeing if waves going in different directions go at the same speed.

In quantum physics, everything moves like a wave and exchanges energy like a particle. So light, for our purposes, acts like a wave. Its speed should not be relative to the speed of its source; it shouldn't act like a bullet fired from a moving car.

But we said that there was no such thing as standing still. That means that there's no special way to move. If light behaved like waves on the surface of water, there would be a special way to move: the way that means the speed of light is the same in all directions.

So according to relativity, light is a wave, but it's a very weird wave: it travels at the same speed in all directions, no matter how fast the person measuring it is going.

For this to be true, something has to be pretty weird, not only about light, but about the universe.

     Why moving clocks run slower 

Let's have an example. The Enterprise D is on impulse drive, flying at half the speed of light towards galactic north. It's about to fly past Deep Space Nine and will pass it 300,000 kilometers away due east of the station. Captain Picard decides to greet Benjamin Sisko by flashing a light, carefully timed to strike the space station at the moment of closest approach. The second Sisko sees the beam, he shines another light back at the Enterprise—or, rather, where the Enterprise will be when the beam arrives.

From Sisko's perspective, the light had to travel along a diagonal path: Picard set off the first light before the moment of closest approach, when the ship was 300,000 kilometers east and some distance south of the station. The second pulse will strike the ship when it is 300,000 kilometers east and some distance north of the station. So the light travels a distance of more than 600,000 kilometers. Since light travels at 300,000 kilometers per second, from Sisko's perspective, more than 2 seconds pass between when Picard sends off his own greeting and when Picard receives Sisko's greeting. (If you do the math—it's not particularly hard math—it comes out to about 2.3 seconds.)

However, from Picard's perspective, the light pulses travel in straight lines: his goes 300,000 kilometers due west, and Sisko's goes 300,000 kilometers due east. The round trip distance is 600,000 kilometers and takes exactly 2 seconds.

So Captain Picard's clock is running slower than Captain Sisko's clock. This is the Time Dilation effect, and is probably the most well-known effect of special relativity—although, as explained above, it tends to be exaggerated in fiction pretty severely.

     Whose clock runs slower? 

So if you are moving, then your clock runs slower.

But remember the first principle of special relativity. If Captain Sisko is on Deep Space 9 and Captain Picard is on the Enterprise, there isn't supposed to be a way to decide that Sisko is the one who's really standing still and Picard is the one who is really moving. If Sisko's clock goes faster than Picard's clock, then that gives us a way to decide that Picard is moving and Sisko isn't, which is exactly what is not supposed to happen under relativity.

Another way to think of this is to run the thought experiment in the last folder again, but this time, have Sisko send the message out first, and have Picard reply to it.

So Sisko and Picard must disagree about whose clock is slowing down.

For simplicity, let's not think about close passes. This time, let's suppose that the Enterprise flies right through Deep Space Nine. Sisko and Picard are in the same place at the same time, so they agree that the Enterprise leaves Deep Space Nine at exactly 4:00. The Enterprise flies away from Deep Space Nine at half the speed of light for one hour (according to Picard's clock), turns around, and comes back. Sisko and Picard compare their watches. Has Sisko seen more or less than two hours pass? It has to be one or the other.

It turns out that Sisko's viewpoint is better than Picard's. Picard had to turn around and come back. That means he had to slow down and then speed up in the opposite direction. That means that he was accelerating, and non-accelerating reference frames like Sisko's are a lot simpler. So since Picard was the one to accelerate, that means that it's his clock that slowed down. So Sisko's watch will say that it's 6:19, not 6:00.

So, to recapitulate, Picard and Sisko will forever disagree about whose clock slowed down. The only way to settle the argument is for Picard to turn around and come back so they can compare notes. The only way to do this is for him to accelerate, and that will pick his viewpoint out as the "wrong" one.

It actually bothered Albert Einstein quite a bit that non-accelerating viewpoints are better. So much, in fact, that he came up with general relativity to explain the physics of acceleration. But that we'll have to save for later.

     Why FTL space ships are time machines 
Let's suppose that the Doctor shows up and offers to help Sisko and Picard decide whose clock is really running slower. Picard leaves Deep Space 9 at 4:00 and flies away at half the speed of light. At 6:19, the TARDIS appears on the bridge, and the Doctor offers to take Picard back to compare notes with Sisko. Picard gets into the TARDIS, and as far as either of them can tell, the TARDIS instantly appears on Deep Space Nine.

When the TARDIS appears on Deep Space 9, Sisko's clock says that it is 6:00. Picard isn't surprised—he knows that Sisko's clock runs slowly, so if the TARDIS traveled instantaneously, then that's exactly what he'd expect to see. Sisko, on the other hand, knows that Picard's clock runs slowly. And he's perfectly entitled to point out that the TARDIS is a time machine, so Picard could easily have traveled backwards in time.

If you accept special relativity, all faster-than-light travel is like this. If you have a faster-than-light spaceship, then from someone's viewpoint, what you really have is a time machine. Even if you only have a Subspace Ansible, so you can talk faster than light but not move faster than light, from someone's viewpoint, you're talking backwards through time. And of course, since all viewpoints are equally valid, you should also be able to travel through time in the ship's own reference frame.

     How to have ships that aren't time machines 

It's possible in fiction to have faster-than-light travel without Time Travel and its attendant problems. You just have to designate one viewpoint as the "right" one: things can move faster than light, as long as they don't travel backwards in time from that viewpoint. This is enough to keep anyone from becoming their own grandparent, since that involves traveling backwards in time from everyone's viewpoint.

This can be justified if you travel via Subspace or Hyperspace. Subspace always has different physics than the real world, so you can decide that subspace does have a correct viewpoint, and so the real world has a correct viewpoint, the one that matches the subspace viewpoint.

It can also be justified if you travel by wormholes: if the wormholes in your universe agree about space and time, you can pick their viewpoint as the "correct" viewpoint.

However, usually FTL with no time travel is simply a case of the writer not doing the research. This tends to be especially jarring in series with both Time Dilation and instantaneous communication, since the logic that leads to Time Dilation and the logic that proves that "instantaneous" means "time travel" are exactly the same.

     Shortening of the way 

Now let's suppose that the Enterprise NCC-1701-D flies past Deep Space Nine and then past Bajor. Bajor is 1 terameternote  away from Deep Space Nine, so it takes light 1 hour to get to Bajor. So from Sisko's perspective, it will take the Enterprise two hours to get to Bajor. But remember, Sisko believes that Picard's clock runs slowly. So Sisko thinks that Picard should think that it takes only 1.7 hours to get to Bajor.

Sisko's got to be right: Picard does think that he passes Bajor 1.7 hours after he passed Deep Space Nine. Why, from Picard's perspective, is this true?

Remember driving down a highway. When you look out a window, it doesn't really look like you are moving forward: it looks like you are sitting still and the trees and buildings next to the highway are moving backward.

Similarly, from Picard's perspective, the Enterprise is holding still, and Bajor is rushing towards it at half the speed of light. It takes 1.7 hours for Bajor to get to the Enterprise, and so from Picard's perspective, Bajor must be only 0.85 terameters away from Deep Space Nine.

This is referred to as "length contraction," or "Lorentz contraction" (after the pre-Einstein theorist who first proposed it), or "shortening of the way". If you think something is moving, then you see it squished relative to the direction of travel. Data and LaForge can measure the Enterprise and conclude that it is 642 meters long, but when the Enterprise flies past Deep Space Nine, Sisko can look out the window and see a ship that seems to be only 550 meters long. And when the Bajoran solar system comes rushing at Picard at half the speed of light, Bajor seems to be only 0.85 terameters from Deep Space Nine.

From Picard's perspective, this is why time dilation happens. Suppose that Picard flies from Deep Space Nine to Bajor and back. From Sisko's perspective, Picard travels 2 terameters at half the speed of light, so it takes 4 hours according to Sisko's clock; since Picard's clock runs slowly, the whole round trip takes 3.4 hours according to Picard's clock. From Picard's perspective, Bajor and Deep Space Nine each travel 1.7 terameters at half the speed of light, so the entire round trip takes only 3.4 hours.

(Since Picard accelerates, describing what he thinks happens to Sisko's clock involves general relativity. See below.)

     Momentum 

Let's suppose that Deep Space Nine and the Enterprise have identical shuttles. Major Kira takes one shuttle and flies it to Bajor. She accelerates to 8% of the speed of light. (At this speed, Time Dilation is going to add about three minutes to the trip, so we're not going to worry about it.)

Meanwhile, the Enterprise is traveling at 86.6% of the speed of light. At this speed, time is dilated and lengths are contracted by a factor of 2. Commander Riker takes his shuttle and does exactly the same thing that Kira did. According to Data and LaForge, Riker is now traveling at 8% of the speed of light relative to the Enterprise. This comes out to 24,000 kilometers per second.

How fast do Sisko, O'Brien, and everyone else on Deep Space Nine think Riker is traveling?

Remember, by length contraction, one of the Enterprise's kilometers is only half of one of Deep Space Nine's kilometers. And by time dilation, one of the Enterprise's seconds lasts for two of Deep Space Nine's seconds. So Sisko and O'Brien see Riker as traveling only 6,000 kilometers per second faster than the Enterprise. They do not think that Riker is traveling at 94.6% of the speed of light (86.6% + 8%). They think that Riker is traveling much slower, only 88.6% of the speed of light.

But Riker did the same thing Kira did. He burned the same amount of rocket fuel.

This means that it takes, in some sense, the same amount of effort to go from 0% to 8% of the speed of light and to go from 86% to 88% of the speed of light. Put another way, the faster you go, the harder it is to speed up.

One way to think about this is that Riker's shuttle weighs more than Kira's—that it gets more massive as you accelerate. This isn't really the best way to think about this, though. Riker doesn't feel any heavier on the shuttle than he did on the Enterprise or back at Deep Space Nine.

A better way to think about it is to simply say that Riker's shuttle, which is going eleven times as fast as Kira's, has 22 times as much momentum, and that increasing its speed by a little bit requires a big change in momentum, and so a lot of fuel.

It turns out that as your velocity gets closer and closer to the speed of light, your momentum grows without bound: if Riker could get his shuttle to the speed of light, then he would have infinite momentum.

Or, to phrase it mathematically:

momentum = mvγ

(At low speeds where γ is about 1, this reduces to "momentum = mv", which is Sir Isaac Newton's definition for momentum.)

     Mass is a kind of energy, or E = mc^ 2 

The equation E=mc2 is an equation, and as such, it cannot really be justified without a bunch of other equations. So in this section, we're going to use the γ from the "Formulas" folder above.

Let's suppose that Kira and Riker take their identical shuttles and dock them at Deep Space Nine. The Enterprise flies past Deep Space Nine at speed v. Riker wants to catch up to the Enterprise, and he doesn't want to burn any fuel to do it, so he persuades Sisko to fire him away from Deep Space Nine at speed v as the Enterprise passes by. Sisko doesn't want to move his station, so he has Kira fly her shuttle away at speed -v. (That is, the same speed, just in the opposite direction.)

Let's suppose that Riker and Kira's shuttles have mass m, and that Deep Space Nine has mass M after the shuttles have been fired away. Before the shuttles have been fired away, the whole agglomeration has mass N. Sir Isaac Newton would expect N=M+2m.

Before the shuttles are launched, from Sisko's perspective, Deep Space Nine has no momentum, because nothing is moving. From Picard's perspective, Deep Space Nine is moving at speed -v, so it has momentum -Nvγ. (Remember that γ depends on v.)

Momentum is conserved. After the shuttles are launched, the shuttles plus Deep Space Nine still have a total momentum of zero from Sisko's point of view.

From Picard's point of view, Riker is now holding still. Deep Space Nine is still traveling with velocity -v. And Kira is now traveling with velocity not quite -2v. Kira's γ is also different from Deep Space Nine's γ.

The point is, from Picard's perspective, Kira's momentum is not -2mvγ. (It turns out to be -2mvγ2.)

Since momentum is conserved from Picard's point of view, we must have that

Nvγ = Mvγ + 2mvγ2.

So when the two shuttles flew away from Deep Space Nine, apparently 2m(γ-1) kilograms just mysteriously vanished.

Now, whatever Sisko (or, rather, O'Brien, Dax and Rom) used to fire Kira and Riker away, it had to have stored energy somehow. Maybe it was an electromagnetic rail gun; maybe it was an enormous spring; but somehow, Deep Space Nine had some amount of stored (potential) energy, that it transformed into the two shuttles' kinetic energy.

You can calculate relativistic kinetic energy just by figuring out how much work it takes to speed up from 0 to v. The answer you get is that it took precisely mc2(γ-1) joules to speed each of the two shuttles up to speed v.

That means that the two shuttles carry kinetic energy 2m(γ-1)c2. And we lost 2m(γ-1) kilograms. That means that every joule of stored energy had a mass of 1/c2 kilograms. Put another way, every kilogram of stored energy gave rise to c2 joules of kinetic energy.

So stored energy has mass, and the energy stored by m kilograms is given by E=mc2.

     History 

In the late 19th century, James Clerk Maxwell published his theorynote  of electromagnetism, which described how electric and magnetic fields interacted. However, as he studied his theory more thoroughly, he found a peculiar result: the equations said that there could exist an electromagnetic field that could "leapfrog" its way through space; an electric field would generate a magnetic field as it disappeared, and this magnetic field would then disappear, and generate the original electric field, which would continue until the continually generating and shrinking fields hit something. The equations also allowed him to calculate exactly how fast the leapfrogging traveled through space, and he was amazed by the result: 299,792,458 metres per second, precisely the speed of light.

However, the physicists were left with a problem, because Maxwell's calculations of the speed of light didn't depend on how fast the measurer was moving; it depended only on the properties of space itself. Did this mean that if Alice was moving when compared to Bob, she would see light moving at the same speed as he did? And if not, why not? Did space change in some way to make it possible?

About the same time there were also experiments by Michelson, Morley, and others to try and detect the lumeniferous aethernote . All of the experiments failed to detect the aether. A fellow by the name of Lorentz came up with some equations which could compensate for what was seen in the experiment.

In 1905, Einstein entered the picture, publishing a paper that said, "Yes, and here is how the universe works because of it." This leads us to the cornerstone of Relativity: the speed of light in vacuum is constant everywhere, and space and time warp to accomodate that.

Einstein's second brainwave it just as important as the first, and arguably more intuitive: Velocity is relative. There is an Urban Legend that he came up with this while sitting on a stationary train, looking out of the window. The train beside his began to pull away, and he realized that his train pulling away would appear exactly the same from the other train, as the other train pulling away appeared to him. (Apart from the acceleration, which will be discussed later.)

General relativity

Objects in the universe don't just move with constant velocities; they can accelerate. You can measure this acceleration without any reference to the outside world. If you were in an elevator with no windows, you couldn't tell how fast you were moving relative to other objects in the universe, but you could tell how fast you were accelerating. If a giant were way way up above your elevator, pulling up on the elevator cable hard enough to accelerate you at 9.8 meters-per-second-per-second, you could measure this acceleration with any number of experiments (including a simple accelerometer you picked up at Radio Shack, or a bathroom scale), even though you couldn't see any objects outside the elevator for reference. You would be pressed down onto the floor of the elevator with a force of 9.8 Newtons for every kilogram of your mass.

But you would not be able to distinguish between the force caused by being accelerated, versus the force caused by sitting in Earth's normal gravity. If your mass were 100 kilograms, and the bathroom scale in your elevator said you weighed 980 Newtons, you could not tell if your 980 Newtons of weight was caused by sitting on the surface of the Earth, or if it was instead caused by a giant pulling up on your elevator cable at 9.8 m/sec2 while your elevator was in deep space.

When Einstein examined this situation, he came to a brilliant conclusion: The force you measure from standing in a gravitational field and not falling is not merely indistinguishable from the force you measure from being accelerated, it was identical to this force. Gravity curves spacetime such that an inertial path through it doesn't follow a straight line; and this curved path through spacetime applies not only to falling objects, but also to light.

     Time again 
In an accelerated reference frame, there's an additional time dilation effect separate and distinct from the time dilation experienced by a fast-moving object. In most cases, the effect is tiny.

The additional factor by which your clock will slow down if you're standing still in the gravitational field of a massive object is:

sqrt (1 - r0/r)

... where r is the distance from the center of the object, and r0 is the "Schwarzchild radius" of the object, equal to 3 kilometers for every solar mass.

For the Earth, the Schwarzchild radius is only 9 millimeters, so a person standing at the surface of the Earth (6000 km away from the center) measures the passage of 0.9999999992 seconds for every 1.0 seconds measured by a stationary observer in deep space. This effect is so tiny it takes atomic clocks to detect.

The effect would become more significant near an extremely dense object, such as a neutron star or a black hole. Say we have a 5 solar mass black hole, which would have a Schwarzchild radius of 15 kilometers. (Incidentally, the Schwarzchild radius for a black hole corresponds to where its event horizon is. Objects that are bigger than their own Schwarzchild radius aren't black holes and don't have event horizons.) Now say you're standing 30 kilometers from the center of the black hole, which is only 15 km above its event horizon. The gravitational time dilation factor here works out to 0.707, which is significant but not huge. For every 7 seconds that elapse for you, 10 seconds elapse for us folks who are a nice safe distance away from the black hole. You'll have to re-synch your clock when you rejoin us — assuming the tidal forces from the black hole don't turn you into molecular spaghetti first — but you're not going to find yourself flung years into the future just by dawdling there for a few hours. If, however, you got closer, such that you were hovering only 1 kilometer above the event horizon, the gravitational time dilation factor would be 0.25, meaning 4 seconds would pass for the outside world for every 1 second you experienced. The effect would get more and more pronounced the closer you got to the event horizon, and when you actually reached the event horizion the gravitational time dilation factor would be exactly zero. You would be frozen in time right at the event horizon for all eternity, as far as the outside world was concerned.

If instead of being stationary, you were in a circular orbit around a massive object, the formula for how much your clock slows down becomes:

sqrt (1 - 1.5 * r0/r)

Note that if your orbital radius r is less than 1.5 times the Schwarzchild radius r0, the expression inside the square root becomes negative. That's because at 1.5 * r0, the speed at which you'd have to be moving to be in a circular orbit is the speed of light. It is physically impossible to orbit any closer than this.

     Light does not go in a straight line 
In order for gravity to work exactly like acceleration, space would have to be curved in the vicinity of any gravitational source. That is, an object following a "straight line" in its own accelerated reference frame would have to follow a path that actually curved around, and the amount of curvature would have to be such that it would end up following exactly the path you'd get if you assumed gravity didn't curve space and you just used Newton's laws instead.

When you assume that a beam of light would follow the same curvature, though, you get a remarkable result. Sir Isaac Newton assumed that light was "corpuscular", that is, made up of little atom-like chunks that were affected by gravity just like ordinary matter, and that the only reason you don't see the beam from a flashlight curving downward on Earth is that light moves so fast that there isn't time for it to "fall" very far. Since we now know the speed of light — which we didn't in Newton's time — you can apply Newton's laws to this hypothetical "corpuscular light" model and deduce exactly how fast a beam of light should fall, or how severely a beam of light should be curved when it passes by a massive object like a planet or the sun. What you actually see when you measure, though, is that the beam of light gets deflected twice as sharply as Newton predicted it should. However, this amount of deflection exactly matches what Einstein predicted would happen if gravity curved space.
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