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Special Relativity Simplified:

January 17, 2014 Astrophysics
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Albert Einstein

Albert Einstein

Einstein put forth special relativity, which explains motion at near-light speeds. Although there are many consequences of Special Relativity, this complex theory consists of only two postulates, both of which are fairly easy to understand. However, to understand these postulates, we must first understand relativity. As an aside, it is a common misconception that relativity came from Einstein, but relativity is an old concept, dating back to Galileo (way back in 1632). Einstein’s Special Relativity, on the other hand, wasn’t published until 1905.

Now then, from his experiments, Galileo deduced that two observers moving at a constant velocity will get the same results from all mechanical experiments. To make it easier to understand, think of it like this: being inside of a vehicle that is traveling at a constant velocity is the same as being inside a vehicle that is at rest (velocity = 0). To prove this, look at where you are right now. You may think that you are at rest, but the Earth is actually moving at an approximately constant velocity through space around the Sun. You don’t feel a thing while sitting there and reading this, and there is no experiment that you can do here on Earth to determine the Earth’s speed.

Hence, one phrase that is important in relativity is the “reference frame.” In relativity, velocity is not absolute, but rather, velocity depends on where you are making the observation. The point at which you are making your observation is where your reference frame is. A reference frame that is moving at a uniform velocity is called an “inertial reference frame.” Ultimately, Galilean relativity can be summarized by saying that all mechanical laws of physics are valid in inertial reference frames. Technically, the Earth is a non-inertial reference frame because of two reasons: 1.) the Earth’s spin on its own axis and 2.) the Earth’s revolution around the Sun. However, the effects of these two are small and, for most purposes, can be neglected.

Another concept of Galilean relativity is the addition of velocities. It can be best understood with this example: you are on a vehicle travelling at 10m/s, then you throw a ball forward at 10m/s. An outside observer will then see the ball to have a velocity of 20m/s. If you threw the ball backwards, the outside observer will then see the ball to have a velocity of 0 i.e., it will appear to stop.

Here’s where Einstein comes in. The first postulate of Einstein generalizes Galilean relativity to include all the laws of physics. Simply stated: “All the laws of physics are valid in all inertial reference frames”. The concept of inertial reference frames is quite easy to understand, but what is a non-inertial reference frame? A non-inertial reference frame is simply an accelerating reference frame. There are “fictitious forces” present in a non-inertial reference frame, the same “force” that pushes us forward when the bus driver suddenly hits the brakes. You may be a bit bored by now, but don’t worry, the more interesting part of Special Relativity is in the second postulate.

Jets of subatomic particles moving at nearly the speed of light via NASA

Jets of subatomic particles moving at nearly the speed of light via NASA

The second postulate states that: “the speed of light in a vacuum is constant in all inertial reference frames.” It simply means that, wherever you look from, you will measure the same speed of light in a vacuum (as long as it moves at a constant velocity). At first, this doesn’t sound so amazing, but when you think of it, it is. For example, if you are in a spaceship travelling at the speed of light and you fire a laser up front, Galilean relativity’s addition of velocities tells us that an observer outside the spaceship will see the laser at twice the speed of light, but Einstein tells us that the observer will still see light travel at light speed. How is that possible? The constancy of the speed of light has two consequences that make it possible, namely, length contraction & time dilation.

Length contraction is the shortening of length of an object travelling at relativistic speeds (speeds that are near the speed of light) relative to an observer outside the object. It is described in a limerick by George Gamow:

There once was a young man named Fisk,
Whose fencing was extremely brisk,
So fast was his action,
The Lorentz contraction,
Foreshortened his foil to a disk.

Time dilation, on the other hand, is the slowing down of time at an object moving at relativistic speeds. This is described in the twin paradox. The paradox starts with two twins who are (of course) the same age. One of the twins (twin A) rode a spaceship that travels at relativistic speeds to some distant place. After traveling for some time, twin A returns again at the same speed. Because twin A moved at relativistic speeds, he would be younger than twin B when he comes home (twin A would have experienced a slowing down of time).

To get an idea of the magnitude of the effects of both time dilation and length contraction, we solve for a magic number, called the Lorentz factor. This number ranges from 0 to infinity depending on the velocity of the moving object. This tells us how much slower time moves or how much shorter the object is relative to an outside observer when moving at velocity v.

Source: Relativity Calculator

Source: Relativity Calculator

At 95% the speed of light, the Lorentz factor will be 3.2. Therefore, for an object moving at 95% the speed of light, time will slow down 3.2 times and the length of the object will be shortened 3.2 times. Moving at relativistic speeds can be seen as time travelling towards the future. If you could move faster than light (although we know that that is impossible), you can move backwards in time instead. An interesting part of Lorentz factor is that it approaches infinity as you approach the speed of light, which means that the effects of time dilation and length contraction increase the faster you move

The reason why we can’t make a spaceship move at the speed of light, or even close to it, can also be seen in the Lorentz factor. In high school physics, we are told that the momentum of an object p is equal to the mass times the velocity, but according to special relativity, the momentum is equal to mass times the velocity times the Lorentz factor. At speeds approaching that of light, the momentum increases. Because of this, it would take infinite energy to move something at the speed of light. Experimental physicists have been able to accelerate particles in large particle accelerators only to very close the speed of light. Don’t worry if what you have learned in high school physics is wrong, the formula for momentum reduces back to p = mv since the speeds that we normally use is much less than the speed of light. This formula would still be correct.

Special relativity is a beautiful theory with stunning implications. Although we can’t observe its effects directly in our everyday lives, by understanding it we appreciate the beauty of the universe more, and this fuels our desire to further explore the secrets of the cosmos.

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