Relativity in brief... or in detail..
Time dilation and length contraction in Special Relativity
Time dilation, length contraction and the relativity of simultaneity are among the strange conclusions of special relativity. This page uses animations to explain them in more detail. There is a little mathematics: we use Pythagoras' theorem about the sides of a right angled triangle, but nothing beyond that.
Now the speed of light c in each of the animations is the same - that's what the principle of relativity is all about. But the distances are different: for Jasper, each tick of Zoe's clock is the time T taken for light to trace the hypotenuse of the right angled triangle. For Zoe, the light pulse covers only w, the width of the car, in each tick. So the constant speed of light means that the light beam clock ticks at different rates for Jasper and for Zoe! Jasper observes Zoe's clock running more slowly than Zoe does. (The red counters give the number of ticks of the clock as measured by each of the observers.) Let's call T' the time that Zoe measures for one tick of the clock, so
Pythagoras meets EinsteinHow to relate, T, the time of a tick on Zoe's clock as Jasper observes it, to T', the time of a tick on the same clock, as Zoe sees it? Look at the triangle at the right of the animation above. Light travels at c, so the hypotenuse has length cT. One side of the triangle is the width w of the car. The other side is vT, where v is the speed of the car relative to Jasper. Pythagoras' theorem says that the square of the hyptenuse is the sum of the squares on the other two sides. Here, this gives
Is time dilation true? How big are the effects?Do clocks at speed really run slower, and do people or things travelling at speed live longer?
One question at a time. Yes, clocks do run more slowly. Planes travel about a million times more slowly than c (so γ is about 1.0000000000005), but atomic clocks are very precise and so this tiny effect effect can actually be measured. In 1971, J. Haefele and R. Keating took atomic clocks on airliners travelling both East (with the Earth rotating underneath them: we could call these "slow frames") and West (these planes have the Earth's rotation speed plus their own, and return to where they came from). Apart from some complications due to the gravitational field variations and their acceleration (which are dealt with by general relativity), this is like the twin paradox, and it gave results in agreement with the relativistic prediction. (See the original paper by J.C. Hafele and R. E. Keating, Science 177, 166 (1972) for details. Also see the diagrams and discussion about this experiment and its complications on the FAQ in high school physics.)
Do people age more slowly? We don't know whether people age more slowly, because even cosmonauts don't travel fast enough for the effect to be statistically observable on their life spans*. However, people's ages are determined by physical and chemical processes in our bodies. Certainly we expect that people would age more slowly at relativistic speeds. Particles certainly do. Particle accelerators generate some short lived particles (eg muons or pions) that travel within a fraction of a percent of c, and (in the laboratory frame) they survive for much longer than their lifetime when at rest in the lab frame. Muons with a half life of 1.5 microseconds are also created several tens of km above the Earth in the upper atmosphere by cosmic rays. Travelling 50 km at c would take 170 microseconds or 110 half lives, so we should expect their numbers to be reduced by a factor of 2110 ~ 1033 (ie effectively none) to reach the surface. In fact they are measured at sea level and at various altitudes, with rates that agree with the relativistic dilation of their half lives. Time dilation happens, however counter-intuitive it may seem at first.
How big are time dilation effects? Note the shape of the curve above: γ only starts to become large at speeds close to c. At 0.99*c, γ is 7. But in many modern devices, electrons are accelerated to higher speeds than this. In a typical electron accelerator used to treat cancers, the electrons have an energy of 20 MeV (see Module 5). The speed of such electrons is 0.9997*c and γ is 40.
Now of course an electron cannot go much faster than this, but it can have a lot more energy. In the Large Electron-Positron collider in Europe's nuclear research lab CERN, electrons (and positrons, or anti-electrons) were accelerated to energies of 100 GeV. For such particles, v = 0.999 999 999 95*c and γ is 200,000. Yes, time is slowed down by that factor. And the momentum is increased by that factor too: something that is rather important in the design of the collider because these electrons must be turned to go in a circle.
Nature can produce even larger particle energies. Some particles striking the Earth's upper atmosphere have energies that exceed 2*1020 eV. If such particles are protons (with mass of about 1 GeV), their speeds would be 0.999 999 999 999 999 999 999 995 c. For them, γ is 1011. Now the age of the universe is about 13 billion years for us, but for such particles, the age of the universe would be about (13 billion years/1011), ie about a month. Such a particle could cross the visible universe in a matter of months (their time).
SimultaneityHaving clocks that run at different rates leads to other strange effects: simultaneity is relative. Whether or not two things are simultaneous depends upon your frame of reference. The time order of events that are close together in time but distant in space can be different in different frames. You have probably noticed this in the preceding animation: in general, Jasper and Zoe do not agree on the time at which the light pulse is reflected.
In the next animation, Jasper has set up an apparatus to make simultaneous 'events'. A little while after he presses the switch, sparks jump at the two gaps, and two pulses of light travel towards him. (Note that the sparks are not simultaneous with the switch: the electric field in the circuit cannot travel faster than light.) The spark gaps are equidistant from Jasper, the light pulses arrive simultaneously, so he concludes that the two events (the two spark emissions) occurred simultaneously. As we'll see later, relative simultaneity can only be noticeable for events that are well separated in space, but close in time. For this reason, we asked Jasper to set up the spark gaps a long way apart, and so as to observe the small time effect, we've slowed everything down in this animation, compared to the previous ones, just for clarity.
Zoe also receives the two pulses of light, which she observes to come towards her at the same* speed c. Jasper has timed the pushing of the switch so that the two pulses, Zoe and Jasper all meet at the centre of the verandah at the same time*. But Zoe sees the light pulses emitted from Jasper's moving verandah, and in her frame of reference the two events (the two flashes of light) are not equidistant. For the two pulses to arrive simultaneously, Zoe deduces that the right hand pulse (emitted from the approaching spark gap) must have been emitted before the emission of the left hand pulse (the receding one), because of their relative motion.
Another non-intuitive result: events that are simultaneous to one observer need not be simulatenous to another. Indeed, the time order may be reversed: a traveller going from right to left with respect to Jasper would, by symmetry, observe the left hand pulse to be emitted first. (A question for you to puzzle on: look at when the switch near Jasper closes and work out why the sparks occur simultaneously for Jasper but not for Zoe. Answer below.)
Sometimes one encounters this objection: in this example, Jasper, sends a pulse of voltage down wires to create his two simultaneous events. What if, instead, he switched two distant switches using two very long, rigid rods. Wouldn't they be simultaneous then? The key word here is "rigid". When you push on the end of a rod of length L, the other end does not move instantaneously. It takes a time L/v, where v is either the speed of sound in the object or the speed of a shock wave in the object. The speed of sound in solids is typically a few km/s. (It is the square root of the ratio of an elastic modulus to the density.) Interatomic forces are continuous functions of separation (see Young's modulus, Hooke's law and material properties), you cannot make an infinitely rigid rod, i.e. you cannot make one with an infinite elastic modulus and thus an infinite v. So, although Jasper might see a mechanical wave travel along each rod at the same speed, Zoe would point out that, to her, the waves have different speeds.
The limits to time order reversalsBut isn't the relativity of simultaneity and time order impossible? Does it mean that some observer can determine that my death happened before I was born?
No. As we show elsewhere, if two events are separated by distance L (according to one observer) and time difference Δt (according to the same observer), then their time order can only be reversed for some other observer if L is greater than cΔt. Let's suppose that a cosmonaut dies 80 years after his birth (Δt = 80 years). For an observer to deduce that he died before he was born, his birth and death would have to be separated in space by more than cΔt, which is 80 light years. But (assuming the cosmonaut is present at both his birth and death - even the busiest people manage to attend both!) to get from the place of his birth to the place of his death, he would have to travel more than 80 light years (L greater than 80 light years) in 80 years. He would have to travel at L/Δt, which is greater than the speed of light, which is in turn impossible. (For quantitative details, see Lorentz transforms, the addition of velocities and spacetime.)
Length contractionYou have probably noticed that, in Jasper's version of events, Zoe's car has shrunk. And vice versa. We haven't proved that yet, but it's logically simple. Suppose that Zoe and Jasper choose to measure lengths in lightyears, lightseconds, lightnanoseconds* etc: ie they measure distance by how long light takes to cover the distance. If they agree on the speed of light, but disagree on measurements of time, they must inevitably disagree on length as well. If you observe someone's clocks run slowly by a factor γ, you will also observe her rulers to be short by a factor of γ: that's the only way that she can measure the speed of light to have the same value you get.
Both agree that the time between the two tags - the time Zoe takes to go past the verandah - is two ticks of Zoe's clock. This is 2T' for Zoe, so the length that Zoe measures is 2vT'. But for Jasper, two ticks of Zoe's clock takes 2T = 2T'γ. The length that Jasper measures for the verandah is 2vT = 2vT'γ. Jasper measures the verandah to be γ times longer than Zoe measures it.
Further, the situation is symmetrical: Jasper observes the car to be shrunk with respect to the verandah, while Zoe concludes that the verandah has shrunk with respect to the car. The proper length is always longer than a measure of the length from another frame. But can't one make a paradox from this? See the "pole in the barn" paradox.
Severe simplifications have been made in the animations shown above. Even if cars could travel at relativistic speeds, this is not how they would "look", because of aberrations associated with the finite time of flight. See References and caveats for more information.
Answer to puzzle. Variations in the electric field in the circuit travels at approximately the speed of light, so the step change in voltage (ΔV) caused by closing the switch travels at this speed to the spark gaps. (For simplicity, we neglected the length of the common part of the circuit - the vertical lines on the diagram - when scaling the animation, because this could be made very short.) For Jasper, this ΔV is travelling (at c) towards two stationary spark gaps. For Zoe, it is travelling (at c) towards two spark gaps moving at 0.8*c. So for Zoe, ΔV arrives at the right hand spark gap first. A caution: although electromagnetic fields, including ΔV, in and around an electric circuit may travel at speeds near that of light, this is not true of electrons. Electron speed in most circuits is only a tiny fraction of c. Exceptions are the electron beams in high energy accelerators.