# Would time stop at the speed of light?

## I am driving my car at the speed of light and I turn on my headlights. What do I see?

Sadly this question and all others about experiences at the speed of light do not have a definitive answer. You cannot go at the speed of light so the question is hypothetical. Hypothetical questions do not have definitive answers. Only massless particles such as photons can go at the speed of light. As a massive object approaches the speed of light the amount of energy needed to accelerate it further increases so that an infinite amount would be needed to reach the speed of light.

Sometimes people persist: What would the world look like in the reference frame of a photon? What does a photon experience? Does space contract to two dimensions at the speed of light? Does time stop for a photon?. . . It is really not possible to make sense of such questions and any attempt to do so is bound to lead to paradoxes. There are no inertial reference frames in which the photon is at rest so it is hopeless to try to imagine what it would be like in one. Photons do not have experiences. There is no sense in saying that time stops when you go at the speed of light. This is not a failing of the theory of relativity. There are no inconsistencies revealed by these questions. They just don’t make sense.

Despite these empty answers, nobody should feel too put down for asking such questions. They are exactly the kind of question that Einstein often asked himself from the age of 16 until he discovered special relativity ten years later. Einstein reported that in 1896 he thought,

*«If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest. However, there seems to be no such thing, whether on the basis of experience or according to Maxwell’s equations. From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. For how, otherwise, should the first observer know, i.e., be able to determine, that he is in a state of fast uniform motion? One sees that in this paradox the germ of the special relativity theory is already contained. Today everyone knows, of course, that all attempts to clarify this paradox satisfactorily were condemned to failure as long as the axiom of the absolute character of time, viz., of a simultaneous, unrecognizedly was anchored in the unconscious. Clearly to recognize this axiom and its arbitrary character really implies already the solution to the problem.»*

In 1905 he realised how it could be that light always goes at the same speed no matter how fast you go. Events that are simultaneous in one reference frame will happen at different times in another that has a velocity relative to the first. Space and time cannot be taken as absolute. On this basis Einstein constructed the theory of special relativity, which has since been well confirmed by experiment.

Questions of relative velocity in relativity can be answered using the velocity subtraction formula *v = (w − u)/(1 − wu/c 2 )* (see relativity FAQ: velocity addition). If you are driving at a speed *u* relative to me and you measure the speed of light in the same direction (*w = c* in my frame), the formula gives *v* the speed of light in your reference frame as, *v = (c − u)/(1 − u/c)*. For any speed *u* less than *c* this gives *v = c* so the speed of light is the same for you. But if *u = c* the formula degenerates to zero divided by zero; a meaningless answer.

If you want to know what happens when you are driving at very nearly the speed of light, an answer can be given. Within your car you observe no unusual effects. You can look at yourself in your mirror which is moving with the car and you will look the same as usual. Looking out of the window is a different matter. The light from your headlights will always go at the speed of light in your reference frame. It will strike any object in its path and be reflected back. Everything else will be coming towards you at nearly the speed of light, so the light reflected from it will be Doppler shifted to very high frequencies—towards the ultraviolet or beyond. If you have a suitable camera you could take a snapshot. The objects passing are contracted in length but because of the different times of passage for the light and effects of aberration, the snapshot will show the objects you pass as rotated. See the relativity FAQ Penrose-Terrell Rotation.

*Ref: Quote from Einstein’s biographical notes in «Albert Einstein, Philosopher Scientist» ed. Schilpp.*

## Why you can’t travel at the speed of light

A lbert Einstein is famous for many things, not least his theories of relativity. The first, the special theory of relativity, was the one that began the physicist’s reputation for tearing apart the classical worldview that had come before. Special relativity, a way of relating the motion of objects in the universe, led scientists to re-evaluate their assumptions about things as fundamental as time and space. And it led to important revelations about the relationship between energy and matter.

Special relativity was published by Einstein in 1905, in a paper titled «On the Electrodynamics of Moving Bodies». He came to it after picking on a conflict he noticed between the equations for electricity and magnetism, which the physicist James Clerk Maxwell had recently developed, and Isaac Newton’s more established laws of motion.

Light, according to Maxwell, was a vibration in the electromagnetic field and it travelled at a constant speed in a vacuum. More than 100 years earlier, Newton had set down his laws of motion and, together with ideas from Galileo Galilei, these showed how the speed of an object would differ depend on who was measuring it and how they were moving relative to the object. A ball you are holding will seem still to you, even when you’re in a moving car. But that ball will seem to be moving to anyone standing on the pavement.

But there was a problem in applying Newton’s laws of motion to light. In Maxwell’s equations, the speed of electromagnetic waves is a constant defined by the properties of the material through which the waves move. There is nothing in there that allows the speed of these waves to be different for different people depending on how they were moving relative to each other. Which is bizarre, if you think about it.

Imagine someone sitting in a stationary train, throwing a ball from where he’s sitting to the opposite wall, a few metres further down the train from him. You, standing on the station platform, measure the speed of the ball at the same value as the person on the train.

Now the train starts to move (in the direction of the ball), and you again measure the speed of the ball. You would rightly calculate it as higher – the initial speed (ie, when the train was at rest) plus the forward speed of the train. On the train, meanwhile, the game-player will notice nothing different. Your two values for the speed of the ball will be different; both correct for your frames of reference.

Replace the ball with light and this calculation goes awry. If the person on the train were shining a light at the opposite wall and measured the speed of the particles of light (photons), you and the passenger would both find that the photons had the same speed at all times. In all cases, the speed of the photons would stay at just under 300,000 kilometres per second, as Maxwell’s equations say they should.

Einstein took this idea – the invariance of the speed of light – as one of his two postulates for the special theory of relativity. The other postulate was that the laws of physics are the same wherever you are, whether on an plane or standing on a country road. But to keep the speed of light constant at all times and for all observers, in special relativity, space and time become stretchy and variable. Time is not absolute, for example. A moving clock ticks more slowly than a stationary one. Travel at the speed of light and, theoretically, the clock would stop altogether.

How much the time dilates can be calculated by the two equations above. On the right, Δt is the time interval between two events as measured by the person they affect. (In our example above, this would be the person in the train.) On the left, Δt’ is the time interval between the same two events but measured by an outside observer in a separate frame of reference (the person on the platform). These two times are related by the Lorentz factor (γ), which in this example is a term that takes into account the velocity (v) of the train relative to the station platform, which is «at rest». In this expression, c is a constant equal to the speed of light in a vacuum.

The length of moving objects also shrink in the direction in which they move. Get to the speed of light (not really possible, but imagine if you could for a moment) and the object’s length would shrink to zero.

The contracted length of a moving object relative to a stationary one can be calculated by dividing the proper length by the Lorentz factor – if it were possible for an object to reach the speed of light its length would shrink to zero.

It is important to note that if you were the person moving faster and faster, you would not notice anything: time would tick normally for you and you would not be squashed in length. But anyone watching you from the celestial station platform would be able to measure the differences, as calculated from the Lorentz factor. However, for everyday objects and everyday speeds, the Lorentz factor will be close to 1 – it is only at speeds close to that of light that the relativistic effects need serious attention.

Another feature that emerges from special relativity is that, as something speeds up, its mass increases compared with its mass at rest, with the mass of the moving object determined by multiplying its rest mass by the Lorentz factor. This increase in relativistic mass makes every extra unit of energy you put into speeding up the object less effective at making it actually move faster.

As the speed of the object increases and starts to reach appreciable fractions of the speed of light (c), the portion of energy going into making the object more massive gets bigger and bigger.

This explains why nothing can travel faster than light – at or near light speed, any extra energy you put into an object does not make it move faster but just increases its mass. Mass and energy are the same thing – this is a profoundly important result. But that is another story.