## 1. The Theory of Relativity

Albert Einstein invented two Theories of Relativity at the beginning of the 20th century, namely the Special Theory of Relativity and the General Theory of Relativity. These theories are now the best ones we have for the description of space, time, and gravity.

### 1.1. The Special Theory of Relativity

The Special Theory of Relativity describes space and time for observers who are not accelerated or decelerated (and hence not rotating). Let's calls such an observer an inertial observer. (Scientists refer to observers in inertial systems rather than to inertial observers.) One inertial observer can have a great speed relative to another inertial observer, as long as neither of them is accelerating or decelerating. Two inertial observers have a constant velocity relative to one another.

The Special Theory of Relativity is based on two postulates (fundamental assumptions):

1. every law of nature has the same form for all inertial observers.
2. the speed of light (in vacuum) is the same for all inertial observers.

It follows from these two postulates that measures of space and time depend on the observer (especially when relative speeds are involved that are close to the speed of light) and that there is no universal inertial observer who has a special place in laws of nature.

### 1.2. The General Theory of Relativity

The General Theory of Relativity is an extension of the Special Theory of Relativity and describes how things are with space and time even when forces act upon the observer. The General Theory of Relativity says that the effects of an acceleration do not depend on the cause of the acceleration (not even if that cause is the force of gravity). It follows that gravity bends rays of light and can slow down time, and that black holes can exist.

## 2. A Journey to α Centauri

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Suppose that one twin stays home and the other twin gets in a rocket and travels to and back from a far-away star at great speed. According to the Theory of Relativity, the travelling twin will be younger than the resident twin, and the difference gets greater when the traveller stays near the speed of light for longer.

For example, if the journey is to the nearest star outside of the Solar System and back (to α Centauri at 4.3 lightyears), if the traveller turns back immediately, and if the rocket has an acceleration or deceleration equal to the acceleration of gravity on Earth (so that the traveller feels the same weight as on Earth), then the greatest speed of the traveller is 0.95 times the speed of light, compared to the Earth-bound twin. According to the resident twin, the whole journey covers 8.6 lightyears and 11.86 years, but according to the travelling twin the journey took only 7.12 years, so at the end of the trip the traveller's age is 4.74 years less than that of the resident twin. The average speed of the traveller was (according to the resident twin) 8.6/11.86 = 0.73 times the speed of light.

It now seems as if the traveller has covered 8.6 lightyears in 7.12 years and hence has travelled at an average speed of 8.6/7.12 = 1.21 times the speed of light, which seems to go against the Theory of Relativity. However, this is a fake problem, because a distance as measured by the resident twin is compared with a time measured by the travelling twin, and the only speeds that really mean something compare a distance and a time as observed by the same observer. Not just the measure of time is relative (and can be different for different observers), but also the measure of space.

According to the Theory of Relativity, a moving object appears shorter in the direction of motion, so the same would hold for a ruler that stretches from the beginning to the end of the journey, and hence also for the distance between the beginning and end. According to the traveller the distance was not 8.6 lightyears but only 4.53 lightyears, covered in 7.12 years, so the average speed of the traveller according to the traveller was 4.53/7.12 = 0.64 times the speed of light.

The Relativistic Travel Calculation Page explains how to make calculations like these.

The Theory of Relativity says that laws of nature look the same for all observers, so when two observers are in similar circumstances then they ought to see similar things, even when they move with respect to one another. The Special Theory of Relativity says that a moving clock seems to run slower than an identical motionless clock, so if observers A and B in space have identical clocks with them and have great speed with respect to each other, then A will see the clock of B run slower than his own, but at the same time B will see the clock of A run slower than his own, too.

For the case of twins of which one (A) stays at home (in space) while the other one (B) travels to a far star and back one could expect that A can say that B travels, but B can just as well say that A travels, so they ought to see similar things, so how can it be that B ages less during the journey than A does (as described above)? This is the so-called Twin Paradox.

The solution of the Twin Paradox is that both observers are not in comparable circumstances, so it is allowed for them to age at different rates. The traveller (B) must accelerate and decelerate long and strongly to get to the far star and back, which the resident twin (A) does not do. The traveller feels the acceleration, but the resident twin feels no acceleration. This is a fundamental difference between the two, so they are not equivalent according to the Theory of Relativity.

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## 4. The Law of Conservation of Mass

In the 18th century, Mr. Lavoisier (1743 - 1794) from France discovered that the total mass of the products of a chemical reaction (whether these products are solids, fluids, or gases) is always equal to the total mass of the ingredients, within the accuracy of measurement. This physical law can be useful for calculations on chemical reactions. In the 20th century nuclear reactions were discovered, and it was found that for such reactions that occur in nature the total mass of the products is less than that of the ingredients. What is going on here?

Atom bombs and hydrogen bombs and nuclear reactors are based on nuclear reactions and not on chemical reactions. During a chemical reaction, the electrons that orbit around the atomic nuclei are distributed differently, but the nuclei themselves remain the same. During a nuclear reaction, the nucleus gets changed (too), which involves much more energy than the redistribution of electrons. That's why it is so much harder to get a nuclear reaction going than to get a chemical reaction going.

Albert Einstein taught us that mass and energy are equivalent and can be transformed into one another. If on balance energy is released or absorbed during a chemical or nuclear reaction, then the total mass of the products of the reaction will be different from the total mass of the ingredients (smaller for net energy release, greater for net energy absorption).

The amount of energy (1 eV) that is involved for a typical chemical corresponds to a mass of about 2 × 10−36 kg, or about one part in a thousand million of the mass of the smallest atom. A mass difference of one part in a thousand million can be measured only with very sensitive equipment and can usually be completely ignored. In a typical laboratory, you can assume that Lavoisier's Law holds. The amount of energy that is involved with a nuclear reaction is much larger. The nuclear reaction from which the Sun derives most of its energy yields about 28 million times as much energy as the typical chemical reaction. Even this is not that much yet, because in this reaction only about one part in 2000 of the mass is transformed into energy. See question 293 for more information about nuclear reactions in the Sun.

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## 5. Mass and Energy

According to Albert Einstein, mass and energy are equivalent, in the sense that mass can in principle be transformed into energy and energy into mass. The Law of Conservation of Energy says that energy cannot be lost and cannot appear out of nothing (if you also take into account energy in the form of mass). Material things have more characteristics than just their corresponding amount of energy, and there are conservation laws for a number of those other characteristics, too. Energy does not possess those other characteristics, so mass can only be transformed into energy and energy into mass if those other laws of conservation can be satisfied, too. All kinds of matter have a non-zero value for at least one of those other characteristics, but energy has zero for all of them.

A pen (for example) can only be completely transformed into energy if it is combined with another material thing that has the exact opposite value in all of those characteristics, so that together they are zero as is required for energy. The stuff that that other thing is made of is called antimatter. If the pen, which is made of ordinary matter, comes in contact with an identical pen made of antimatter, then both of them would transform into an enormous amount of energy. The other way around, a certain amount of energy can be transformed into equal amounts of ordinary matter and antimatter (but it is very unlikely that that would be in the form of a pen and an anti-pen).

One of the big riddles of science today is why there is matter in the Universe but hardly any antimatter. If the Universe was formed out of energy (which fits current theories), then there ought to have been formed equal amounts of matter and antimatter (because of the conservation laws), but it turns out that more matter than antimatter was created, at about one particle in a thousand million. Fairly soon after the formation of the Universe, all antimatter particles teamed up with a corresponding matter particle and transformed into energy again, except for those one-in-a-thousand-million matter particles for which no corresponding antimatter particles were around, and those matter particles make up all of the mass in the Universe today. It may be that the conservation laws can be broken at the enormously high temperatures and pressures that prevailed at the beginning of the Universe, but it is very difficult to check that today, because we cannot recreate those circumstances in our laboratories.

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In daily life, you can add up speeds in the same direction. If you are in a train that goes 100 km/h relative to the ground, and if you run at 10 km/h relative to the carriage in the direction that the train is going, then you go at 100 plus 10 or 110 km/h relative to the ground. But if the speeds get close to the speed of light, then this simple addition no longer holds.

One of the foundations of the Special Theory of Relativity is that everybody measures the same value for the speed of every ray of light in a vacuum, compared to the measuring device. If we on Earth measure how fast sunlight goes compared to us, then we measure the speed of light, commonly denoted c. If at the same time a rocket flies past the Earth at 0.8 c (0.8 times the speed of light) toward the Sun, then a simple addition would suggest that the sunlight travels at a speed of c + 0.8 c = 1.8 c compared to the rocket, but if an astronaut in the rocket measures the speed of sunlight compared to himself, then he'll instead find just the speed of light, c!

The formula with which speeds in the same direction add up in the Special Theory of Relativity is:

$$w = \frac{u + v}{1 + \frac{u v}{c^2}}$$

In this formula, $$u$$ and $$v$$ are the two speeds you want to add, $$c$$ is the speed of light, and $$w$$ is the sum of the speeds.

If $$u$$ and $$v$$ are both much smaller than $$c$$, then the results of this formula are almost the same as when you just add $$u$$ and $$v$$. For example, if $$u$$ and $$v$$ are both equal to 1000 kilometers an hour (the speed of a jet plane), then $$w$$ is equal to 2000 kilometers an hour minus 0.0000017 meters an hour.

If $$u$$ or $$v$$ or both are equal to $$c$$, then the result is also equal to $$c$$, and if $$u$$ and $$v$$ are both less than $$c$$, then the result is also less than $$c$$.

For example, if $$u = 200,000$$ km/s and $$v = 100,000$$ km/s, and (for convenience) $$c = 300,000$$ km/s, then the sum is

\begin{align*} w & = \frac{300 000}{1 + \frac{200 000×100 000}{300 000^2}} \\ & = \frac{300 000}{1 + \frac{2}{9}} = 300 000×\frac{9}{11} = 245 455 \text{ km/s} \end{align*}

which is clearly below the speed of light.

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## 7. Relativistic Rotation

A relativistic speed is a speed that is close to the speed of light, so that classical theories that provide accurate predictions for lower speeds are no longer good enough, and instead the more complicated Theories of Relativity must be used.

The only celestial objects that I know about that might have rotation speeds close to the speed of light are neutron stars (such as pulsars). A neutron star with a radius of 10 km that rotates at a rate of 640 times per second (the fastest rate I've seen reported for one) would have an equatorial rotation speed of 0.13 times the speed of light.

Ordinary stars and clouds of gas rotate at speeds very much less than the speed of light. Rotation (angular momentum) is a conserved quantity, so in practice you can only get an object such as a star or a cloud of gas to rotate much faster by decreasing its radius greatly, just like an ice skater rotating around her axis rotates slower if she extends her arms and faster if she holds them close to her body. That is how a neutron star is formed out of a very massive ordinary star: The material in the center of the star is squeezed into a very much denser state, so it takes much less space, so the star's core shrinks to a much smaller size and its rotation speeds up as a result.

Something in or on a rotating body does not notice the rotation speed, but only the balance of forces acting on it. Someone on such a body won't notice any fundamentally relativistic effects in its local area associated with just the rotation speed.

If you could steadily increase the rotation rate of whatever celestial body you are on the equator of, then you'd notice yourself getting steadily lighter, because the increasing centrifugal force would balance ever more of the force of gravity pulling you down. If the rotation rate grew too large, then the centrifugal force would equal the force of gravity, and then you would float. I expect that the celestial body would start disintegrating soon after, if you kept increasing the rotation rate.

You'll notice that the speed of light did not feature in this description.

For bodies that are far from turning into a black hole, the rotation speed at which the body starts disintegrating is much less than the speed of light. Only bodies that are close to turning into a black hole (i.e., whose radius is not much greater than the Schwarzschild radius appropriate for their mass) can have rotation speeds close to the speed of light (i.e., relativistic speeds).

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## 8. Travelling through Time

You can (at least in your imagination) travel through time by changing the time (which we'll call time travel of the first kind) or by changing the speed at which time flows (time travel of the second kind).

Time travel of the first kind lets you go to a time in the past or future without needing to pass through all of the times in between. Such time travel is not possible, as far as we know, except perhaps with the help of exotic things such as "worm holes", of which it is not yet clear if they (can) exist.

Time travel of the second kind works by making your time flow at a different rate than the time of someone else, so that your time keeps running slow or fast compared to the time of the other person. In this way you can slowly accumulate a time difference with others. Time travel of the second kind is possible, because there is no single universal clock that ticks at the same rate for everyone. How long something seems to be or to last depends on the circumstances of the observer, even if all of the measurements are accurate.

That there is no universal clock that runs at the same rate for everyone was proposed by Albert Einstein at the beginning of the 20th century to explain a number of strange observations, such as the observation that the speed of light in empty space is always the same, regardless of the speed of the sender or the receiver of the light. This is very different from how things work in ordinary life.

If you throw a ball forward from a moving car then that ball goes faster relative to the ground than if you throw it from a car that is not moving. If you throw the ball backward from a moving car, then it goes slower relative to the ground than if you throw it from a stationary car. This effect does not hold for light. Whether you shine a light forward or backward from a moving car, the light always goes equally fast relative to the ground (and also relative to you in the car!). You cannot tell how fast the sender moves from the speed at which light reaches you. If distance and time were independent and absolute, then you would be able to tell something about the speed of the sender from the speed of the received signal, so distance and time are apparently not universal.

### 8.1. Time Dilation

It follows from Einstein's Theory of Relativity (which explains the strange observations) that if you keep changing your speed for some time while your friend doesn't move, then your clock goes slower than the clock of your friend. This is called time dilation. If you make a long journey through space in a fast rocket, then your journey will take less time as measured by your accurate ship's clock than as measured by an identical clock back at home. This effect is only noticeable when speeds are reached that are close to the speed of light. In your daily life, the effect is very small. For example, if you accelerate from 0 to 100 km/h (62 mph) in 20 seconds and then slow down to 0 again in another 20 seconds, then you'll have remained 20 femtoseconds (0.000,000,000,000,06 seconds) younger relative to your friend who didn't move, but both of you have of course aged by about 40 seconds.

The Theory of Relativity also says that clocks in stronger gravity run slower than clocks in weaker gravity, but that effect is only very small in daily life as well.

The corresponding formula is

$$\frac{∆t}{t} = \frac{∆Φ}{c^2} \label{eq:dtt}$$

where $$∆t$$ is the time difference between two otherwise identical clocks, $$t$$ is the time between the two measurements (for both clocks), $$∆Φ$$ is the gravitational potential difference, and $$c = 299792458 \text{ m/s}$$ is the speed of light in a vacuum. Formula \eqref{eq:dtt} is only valid if $$∆t$$ is much smaller than $$t$$.

The gravitational potential outside of a spherically symmetric planet is equal to

$$Φ = -\frac{GM}{r}$$

where $$G = 6.672×10^{−11}$$ Nm²/kg² is the universal gravitational constant, $$M$$ is the mass of the planet, and $$r$$ is the distance from the center of the planet.

If clock 2 is $$∆r$$ higher (in altitude) than clock 1 (with $$∆r$$ much smaller than $$r$$), then

$$∆Φ = \frac{GM∆r}{r^2} = g∆r$$

where

$$g = \frac{GM}{r^2}$$

is the gravitational acceleration. In that case,

$$\frac{∆t}{t} = \frac{∆r g}{c^2}$$

If the two clocks are not far from the surface of the Earth, then $$g ≈ 9.81$$ m/s². If $$r$$ is measured in meters, then

$$∆t ≈ 1.09×10^{−16} t ∆r \label{eq:opp}$$

If the second clock is at 1000 m higher altitude than the first clock, then $$∆r$$ = 1000 m. If we look after a year how far the clocks have gotten out of step, then $$t$$ = 1 year = 3.156 × 107 s, and then formula \eqref{eq:opp} yields $$∆t$$ ≈ 1.09 × 10−16 × 3.156 × 107 × 1000 = 3.4 × 10−6 s. After a year the clocks are running 0.0000034 seconds out of step.

If clock 1 is near the Earth (at distance $$r$$ from the center) and clock 2 is in space far from Earth but at the same distance from the Sun as the Earth is, then

$$∆Φ = \frac{GM}{r}$$

so

$$∆t ≈ \frac{GMt}{rc^2}$$

If clock 1 is on Earth, then $$r$$ = 6378 km and $$M$$ = 5.976 × 1024 kg, so then

$$∆t ≈ 7.0×10^{−10} t \label{eq:dt2}$$

If clock 1 is on Earth and clock 2 is far in space, and if we look after a year how far they have gotten out of step, then $$t$$ = 1 year = 3.156 × 107 s and then formula \eqref{eq:dt2} yields $$∆t$$ = 7.0 × 10−10 × 3.156 × 107 = 0.022 s. After a year the clocks have gotten 0.022 seconds out of step.

So you can in fact travel through time, in a sense, but only into the future, and only by passing through all intervening moments, and (if you wish to accumulate an appreciable time difference) only by travelling very far through space. Such time travel does not seem very useful, because you'll not be able to go back into time afterwards to tell everyone what it was like in the future. In a time travel station, you'll be able to buy only one-way tickets to the future.

That this kind of time travel is possible is because the Theory of Relativity turns out to be valid in our Univese. Why this should be the case is a question to which science does not have an answer (at present).

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### 8.2. A Star's Life in a Few Seconds?

Even the most massive and therefore brightest and shortest-lived stars take a few million years to complete their life cycle, as measured by an accurate clock at rest compared to the star. Through time dilation (explained above) you could, in principle, make the lifetime of such a star seem to take only a few seconds by your clock, but you cannot make it so that it appears to take the same amount of time for everybody else, too. In practice, it is vastly beyond our capabilities to reach circumstances where you'd see the evolution of a star proceed at a rate very different from its proper rate (the rate as measured by a clock at rest compared to the star). Also, those circumstances would be bad for your health.

For example, if you could get into circumstances where one million years of the life of the star would seem as one second to you, then you'd receive in that one second all of the energy from that star that you'd receive spread out over one million years under ordinary circumstances, and that energy would arrive mostly in the form of very powerful gamma rays rather than as visible light.

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