###### By Robert M. Hazen, Ph.D., George Mason University

## According to Maxwell’s equation the speed of light was a universal constant. However, the behavior of light in different frames of references led to a paradox: How could the speed of light be constant, even for an object traveling towards you or away from you at near the speed of light? How did Einstein connect light and relativity?

### Paradox of light Being a Constant

Maxwell’s equation claims that the speed of light is a universal constant, 186,000 miles per second so long as that speed of light is measured in a vacuum. But this statement leads to a paradox, as velocities are additive in our everyday experience.

But, then think about the point of view of a space traveler who’s sitting at a stationary position in space looking at the Earth. We, on Earth experience complex circular motions at the same time, so what is the proper frame of reference?

Most physicists would argue that all frames of reference are equal in the sense that you can use them to determine the natural laws, and those natural laws should be the same. This assumption, however, leads to a paradox when it comes to the behavior of light. If one stands at some place and shine a flashlight, that light travels at 186,000 miles per second.

Now, according to Maxwell, it doesn’t matter whether the object is moving towards you, or away from you, or whatever, that’s the speed of light. So, according to Maxwell’s equation, the velocity of light is not additive, it’s an absolute constant.

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### Einstein’s Attempts to Solve the Paradox

If light speed is constant, then velocities are not always additive. Albert Einstein realized that there were really only three possible ways to resolve the dilemma.

You could say that the laws of nature are different in different frames of references; so that a different version of Maxwell’s equation is required in different situations. There was the possibility that Maxwell’s equations could be wrong. Perhaps the speed of light is additive in a normal way, and that the equations of Maxwell just don’t take into account this property of light.

### Reasons for Reluctance

Einstein and his contemporaries were quite reluctant, however, to accept this idea because everything else about Maxwell’s equations described electricity and magnetism so perfectly it seemed to be a complete and thorough mathematical framework, and so why should light violate that framework?

And then there was another possibility. Our common sense about the way velocities add could be wrong. Our everyday experience doesn’t deal with objects traveling 186,000 miles per second.

We can’t perceive that kind of velocity. So, we don’t have any everyday experience, and hence maybe our intuition is just plain wrong. Einstein chose to accept this case, but when he did, he found out that the consequences quite bizarre.

Learn more about Newton’s laws of motion.

**Theory of Relativity**

The principle of relativity states that Newton’s laws of nature are the same in every possible reference frame. In other words, every observer, anywhere in the world, anywhere in the universe sees the exact same laws of nature. So Einstein’s theory of relativity begins with this assumption and considers logical consequences of light having the same speed in all different reference frames.

The first part of the theory of relativity was published in 1905. That’s special relativity, and it deals with reference frames that are not accelerating, that are in uniform motion. Remember, Newton divided all motion into uniform and accelerating motion. And then the general relativity published much later, in 1916. Mathematically much more complex, and that dealt with accelerating reference frames.

### How did Einstein Connect Light and Relativity?

Here is the story about how Einstein began thinking about light and relativity. He was riding on a trolley car in Bern, Switzerland, and he looked out the window at a clock tower. And just as that clock tower was about to strike noon, he imagined what would happen if the trolley car sped up to almost the speed of light.

Here’s what Einstein said would happen. He would be surfing on a wave of electromagnetic radiation coming from that clock. But the clock from which he was receding at the speed of light, he’d be riding the crest of the wave that carried the information that it was 12:00 noon. The clock would have appeared to stop because the information that it’s 12:00 noon is traveling away from the clock at the same speed that you’re traveling away from the clock.

So what one perceives as the passage of a second, or a minute, or an hour, or a day, might be quite different from what someone else perceives as the passage if we’re traveling at very different velocities.

### Time Dilation

Let’s take an example. Set up a light clock that was 186,000 miles long, and that clock would have a tick and a tock one second, two seconds, one second, two seconds, because light travels 186,000 miles per second. So, if you’re standing on the light clock, you’d just see seconds ticking away normally.

Now imagine if that clock is moving relative to you. In that case, you’d see the light pulse leave, the light flash, and the light has to adopt an angular path, and then the light goes on an angular path again, angular path.

The path that the light is traveling as you observe it is much longer. And because the path is longer, it need not be just 186,000 miles; it could be twice that long. The clock appears to slow down because the light has to be traveling 186,000 miles per second, and yet it appears to have to travel a farther distance.

Learn more about universal gravitation.

### The Lorentz Equation

The exact amount of time dilation can be calculated using simple geometrical arguments. Einstein did this calculation, and it turns out that every time a pocket watch ticks one second, the light clock moving at a velocity of *v*, ticks only a fraction of a second, and that fraction of a second can be given by the Lorentz equation.

Lorentz equation says that the time, which is some fraction equal to or less than 1, is the square root of [1 minus the quantity (v/c)^{2}], where v is the velocity in which you observe the clock moving and c is the speed of light.

### Common Questions about **the Theory of Relativity**

**the Theory of Relativity**

**Q: Define Lorentz equation mathematically?**

Lorentz equation states that the time, which is some fraction equal to or less than 1, is the square root of [1 minus the quantity (v/c)2], where *v* is the velocity in which you observe the clock moving and *c* is the speed of light.

**Q: What is the underlying principle of relativity?**

The underlying principle of relativity is that the laws of nature are independent of the observer’s frame of reference.

**Q: When were the theories of relativity published?**

The first part of the theory of relativity was published in 1905, and the second in 1916.