Special relativity

The theory of special relativity was developed by Albert Einstein, and published in 1905. It describes the behavior of objects traveling at very high speeds - close to the speed of light in vacuum - with respect to other objects. A prediction of the theory is that when the relative velocity of two objects is an appreciable fraction of light's speed, time passes at different rates for those objects. Distances also vary; for example, length shortens in the direction of motion.

An important postulate of the theory is that the speed of light in vacuum always is constant (its value is denoted by the symbol c by physicists, and by c₀ in the SI system of units), no matter what the relative speed of the object emitting the light to that of the object receiving it. Unlike the 'classical' example of a bullet fired from a train (where the bullet's speed relative to the ground, or another train, depends on the muzzle velocity of the gun, and speed of the trains), the speed of a beam of light is always c, no matter what the velocity of the source, and the device measuring its speed.

These proposals utterly contradicted humans' intuitive view of the everyday world; in particular, our perception of time and distance, while quite correct in everyday life, improperly extends these intuitive ideas to high speeds. Because our intuitive understanding ultimately misunderstands them fundamentally, human perception fails to grasp what are thought to be the true nature of time and distance.

The theory was in accord with the paradoxical results of certain 19th century physical experiments which attempted to detect the universe's background ether, which was supposed to be the ultimate neutral background, or reference framework, against which the entire physical universe moved. Physicists had always assumed the ether's existence, but experiments--most notably the Michelson-Morley experiment of the 1880s--always failed to detect it.

By boldly refusing to assume the possibility of an ether and theorizing laws of motion without referring to an absolute background, Einstein's simple presumption of objects' "relativity" revolutionized the fundamental view of the physical universe. Astonishingly, Einstein developed the theory when he was only a twenty-six year old clerk in the Swiss patent office.

Einstein's assumptions

Einstein rested his theory on two postulates. He presumed physical experiments performed in any room moving at any constant speed in any constant direction, that is, in any inertial frame, must always produce the same results. In other words, all physical laws should take the same form in all inertial frames, including the laws of electromagnetism. This notion is shared by Newtonian mechanics and the mechanics of Galileo, but those formulations considered only mechanical behavior because electromagnetism was not yet extant. Along with this proposal, called by Einstein the Special principle of relativity, Einstein proposed as well that the speed of light in vacuum should be the same for all inertial observers, which is the key postulate separating special relativity from earlier mechanics. This postulate was in accord with work published by experimental physicist Albert Michelson in 1881 and with greater accuracy in collaboration with Edward Morley in 1887, although these experiments may not have been a strong influence upon Einstein at the time. The Michelson-Morley experiment aimed to determine the speed of light relative to the background ether, which required detecting differences in light's speed depending on how it moved through that ether. Surprisingly, the experiment found that light moves with exactly the same speed all the time, regardless of the motion of the object from which the light emanates or is measured.

Aside from its basis in physicists' experimental results, assuming the constancy of light's speed also does not contradict human perception in any obvious way. In everyday life we experience light's speed as invariably infinite: turn on a light switch and a room is illuminated instantaneously. A simple thought experiment, however, reveals the strangeness of light's speed:

Imagine driving a car straight down a highway at 60 mph. An observer on the side of the road measures our speed at 60 mph. If another car comes toward us at 50 mph as measured by the observer on the side of the road, we inside our car would perceive it coming at us at (60 + 50) = 110 mph. Both cars and the outside observer are in inertial frames. From experience, we know that speeds simply add together. Now imagine that we turn on our headlights. Designating the speed of light in the traditional manner by the symbol c, we see the light beam travel away from us at light's constant speed c. We might also presume that the oncoming car's driver sees our light beam traveling at (c + 110) mph because experience tells us we must add the speed of our car and the oncoming car to the speed of our light beam. Our assumption that observers always measure light's speed as the same, however, means that the other car sees the light beam moving at speed c as well, and that the extra 110 mph makes no difference. The observer on the side of the road must also see our light beam traveling at speed c even though it emanates from a moving car. The cars' speeds make no difference. If both cars were traveling at half the speed of light, the oncoming car would still measure our light beam as traveling at speed c, not at c + (1/2) c + (1/2) c, regardless of the great speed of the two cars.

Time dilation

Consider another thought experiment. We travel in a train uniformly in one direction at speed v (that is, we're in an inertial frame). An observer stands motionless on the side of the tracks watching our train coach pass (that is, he's also in an inertial frame). We shoot a light beam from a flashlight straight up at a mirror on the train coach's ceiling a distance h from the flashlight. Inside the train we see the light beam go straight up, hit the mirror, and come straight back down, covering the distance h twice, which is a total distance of 2h. Let's call the short amount of time this experiment takes t' . During time t' the train travels a short distance.

(PD) Image: John R. Brews
Top: View of the light beam's path from inside the train. Bottom: View of the light beam's path from outside the train.

Now consider what the observer on the side of the tracks sees. Because the train moves, he does not see the light beam go straight up and down, but sees it climb at an angle, hit the mirror, then travel back down at the same angle to hit the flashlight which has now moved a short distance L. It is worth noting that while the observer on the train can use the same clock to register departure and return, the observer on the track must use two clocks because the departure and return of the light occur at two separated locations.

On the track, each leg of the light beam's journey is a distance greater than h, so the light has traveled a distance greater than the 2h we measured inside the train. To make this point, imagine a line coming down from the mirror forming a right triangle one of whose sides is the mirror's height h and the other side half the distance traveled, say L/2. By the Pythagorean theorem, the hypotenuse must be greater than h, in fact √(h²+(L/2)²), so the total distance traveled must be greater from the track observer's point of view.

Let's call the total distance we saw the light beam travel aboard the train d and the track observer's greater distance D. According to the track observer, suppose the time the experiment took on the train is t' , but for those on the track it is t. The light beam's speed is the same on or off the train, c.

Since distance = rate × time we now have

$d=ct^{\prime }$

and

$D=ct\ .$

Because d < D and c is constant, we must conclude that t' < t, that is, from the track observer's viewpoint, the train observers see a shorter time interval than is seen from the track.

The time relationship is made quantitative using the triangles in the lower panel of the figure. We note that the height of the right triangle is h=ct' /2, the base is vt/2 and the hypotenuse is ct/2, so using Pythagoras theorem:

$\left(ct/2\right)^{2}=\left(ct^{\prime }/2\right)^{2}+\left(vt/2\right)^{2}\ ,$

or,

$t=t^{\prime }\ {\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\ ,$

with t the time on the track, and t' the time on the train. Evidently, the square root is smaller than 1 and so t >t' , as previously concluded.

Of course, the same experiment can be done on the track. Then the light goes vertically up and down according to the track observer, who uses one clock, but follows a triangular path as seen by us on the train using two clocks. By the same reasoning, to us on the train the observers on the track observe a shorter time interval than we do on the train. Thus, the situation is symmetric, and whichever observer is perceived as performing the experiment while apparently moving also is perceived to measure a shorter time interval between the events.

This example can be converted to a comparison of clocks. Let's use the flashlight and mirror as a clock. Suppose that a unit of time is that for the light to go from the flashlight to the mirror and back. The same clock construction is used on the train and on the track. Both the clock on the track and that on the train consider one unit of time to be 2h/c, and h and c are the same for both. But to those on the track watching the traveling clock, the traveling clock on the train counts one unit of time as the longer time Δt' = 2√(h²+(L/2)²)/c. Thus, the experiment of watching the clock on the train tick a unit of time takes a time Δt' according to those on the track.

On the track, two clocks are used to observe the single clock on the train. To synchronize the two clocks, a light signal is sent to the distant clock and the distant clock is synchronized by setting it so that when the light arrives the time there is L/c, where L = vΔt' is the clock separation, and Δt' is the time of the experiment (according to those on the track). The height h on the other hand is h=cΔt/2 according to their clock. Substituting:

$\Delta t^{\prime }={\frac {1}{c}}{\sqrt {(2h)^{2}+L^{2}}}={\frac {1}{c}}{\sqrt {(c\Delta t)^{2}+(v\Delta t^{\prime })^{2}}}$

$={\frac {\Delta t}{\sqrt {1-v^{2}/c^{2}}}}\ ,$

which is to say the clock on the train appears to run slower than the clock on the track because the unit of time on the train appears longer. Because the unit of time is longer on the train according to those on the track, the separation in time of two events as measured on the train appears to be shorter than is measured on the track, which is how this example started out.

Again, the same comparison of time units can be done from the train observer's viewpoint, for whom the clock on the track is moving. For the train observer, the clock on the track runs slower.

This slow-down of clocks perceived to be moving is called time dilation.

Lorentz contraction

Let the length of the coach be L₀ as measured by those on the train by sending a light signal from one end of the car to the other and reflecting it back:

$L_{0}=c\tau /2\ ,$

where τ is the time of the round trip. For those on the track, this time is too short because moving clocks run slow, and on the track this time is seen as τ/√(1-v²/c²). But the time seen by those on the track can be calculated as follows: let the time for the outward signal to reach the mirror be t₁. Then the outward light signal travels a distance L + vt₁ as the mirror retreats from the source a distance vt₁, where L is length of the coach according to those on the track. Accordingly:

$t_{1}=(L+vt_{1})/c\ ,$

or, solving for the time:

$t_{1}={\frac {L/c}{1-v/c}}\ .$

In the same way, the return light signal travels a distance L − vt₂ as the receiving point moves toward the return signal an amount vt₂. The time t₂ is then:

$t_{2}={\frac {L/c}{1+v/c}}\ .$

Adding the times produces:

$t_{1}+t_{2}={\frac {2L/c}{1-v^{2}/c^{2}}}={\frac {\tau }{\sqrt {1-v^{2}/c^{2}}}}\ ,$

or:

$L={\frac {c\tau }{2}}{\sqrt {1-v^{2}/c^{2}}}=L_{0}{\sqrt {1-v^{2}/c^{2}}}\ .$

In words, the length seen on the track is smaller than the length seen on the train, a phenomenon called Lorentz contraction. The frame where the coach appears to be stationary (that is, the frame of the moving train) is called the rest frame of the coach because it doesn't move in this frame, so to restate matters, the Lorentz contraction always makes the dimension of a moving object measured in the direction of its motion less than that dimension in its rest frame.

Lorentz transformation

Time dilation specifies that time passes more slowly in an inertial frame that moves relative to our own, countering human perception of time's universally uniform passage. Time's passage, however, differs infinitesimally even well beyond the top speeds humans can achieve with the help of technology such as rocket propulsion. To calculate the magnitude of time dilation and the relativistic effect on length in the direction of motion, Special Relativity employs the Lorentz transformation first proposed by Hendrik Antoon Lorentz.

To compare common-sense with the Lorentz transformation, consider first a common sense scenario. Suppose that we are motionless with our clock in an inertial frame (the reference frame) as a car with its own clock passes by in its inertial frame (the primed frame) at velocity v. The car's length, which we measured beforehand as L, is measured by its driver in the car's inertial frame as L' . If, according to our clock, we time an interval t after the car passes us, we know from experience the car's clock will also have passed the same time t. If we measure the car's length as it goes by, our common sense and measuring technique also gives us length L. The formulas t' = t and L' = L follow from the Galilean transformation.

As we have already seen, though, Special Relativity's effects elude our common-sense perception, so to calculate the change in time and length we now use formulas following from the Lorentz transformation:

$t^{\prime }={\frac {t}{\sqrt {1-{v^{2}}/{c^{2}}}}}$

and

$L^{\prime }=L\ {\sqrt {1-{v^{2}}/{c^{2}}}}$

Generally speaking, the Lorentz and Galilean transformations specify two different mathematical techniques for mapping one Cartesian coordinate system onto another moving (primed) Cartesian system. We only apply them here to tell us how uniform motion relativistically induces time dilation and length contraction, also called Lorentz contraction.

For human scaled velocities – always infinitesimal relative to the speed of light – the fraction (v²/c²) is so close to zero that the quantity under the square root in both formulas is effectively 1. Since the square root of 1 is 1, we can see these formulas reduce to those conforming to common sense, which tells us time passes uniformly and object's length doesn't vary just from an object's uniform velocity, in accord with the formulas from the Galilean transformation. Should v, however, become an appreciable fraction of c, v²/c² gets closer to 1 so √(1 - v²/c²) gets close to zero. This implies that L' becomes only a small fraction of L, that is, that length contracts in the primed frame. In the time formula, we invert this same fraction (so that it's now greater than 1) and move it to the other side of the equation to see that t' must be multiplied by some number greater than 1 in order to equal t. This means more time t passes than time t' , so time passes more slowly in the primed frame.

The twin paradox

Notice that the principle of relativity allows us to say that from the observer inside the car's point of view, he is at rest and it is the observer on the side of the road who moves. Since both observers are in inertial frames, physical experiments must always produce the same results, namely that the car's driver observes our time pass more slowly than his own. How is it possible for both frames to see the other as passing more slowly? Couldn't they stop, meet, and determine whose clock shows greater time passage? It could only be one clock or the other.

This is an example of the well known twin paradox:

According to those on Earth, a twin taking a space trip at high speeds has a slower biological clock than an Earth-bound twin. The traveler therefore returns to Earth to find their stay-at-home sibling has aged in comparison. The age difference seems a paradox if one adopts the view that, to the traveler, the Earth-bound sibling appears to experience a high speed history, and so should age more slowly according to the traveler. That is, to the traveler upon return, the stay-at-home should be the younger twin, contradicting the Earth observers' expectations.

The details in resolving the paradox go beyond this article's scope. Suffice it to say that the twins are subject to relative acceleration and thus do not remain at all times in two frames that are related as inertial frames. Acceleration implies departure from an inertial frame, and Special Relativity regards only the laws of physics in inertial frames. Once forces like acceleration (or deceleration) or gravity are introduced, one must turn to the Theory of General Relativity to explain motion's effects on length and the passage of time.