Yesterday we looked at the two postulates of Einstein’s Theory of Special Relativity. We saw how our view of time is changed by taking the two postulates and applying them to motion at or near the speed of light. Now let’s look at mass and acceleration at light speed.
Looking at the top equation, which we presented yesterday, you can see that at the speed of light, the equation’s denominator becomes 1 – 1, which is zero. Time stops. If the velocity could somehow exceed the velocity of light, the denominator would be the square root of a negative number, which is not possible.
Another one of Einstein’s equations is a description of length in the direction of motion. The second equation shows that an object’s length in motion (L’) is equal to its length at rest (L) times the quantity square root of 1 minus the velocity (v) squared divided by the speed of light (c) squared. Thus the faster you move, the thinner you are in the direction of motion. An object one meter long at rest would be .765 meters at half the speed of light. At the speed of light, it would disappear, because it would have no length. There would be energy, but no physical length. What travels at the speed of light? The answer, of course, is light itself. Light is two-dimensional. It has no thickness in the direction in which it is moving, precisely what Einstein’s postulates predict.
Mass is another quantity that is affected by Einstein’s postulates. The equation for mass is similar to the equation for time. The mass in motion (m) equals the mass at rest (m’) divided by the square root of 1 minus the velocity squared divided by the speed of light squared. As an object moves faster, its mass increases, but it can never reach light speed. What, then, can we know about mass and acceleration at light speed?
One of the fundamental laws of physics is Newton’s Second Law. It says that when we apply force to a mass, the force (F) depends on the amount of the mass (M) and how much we want it to accelerate (A). The equation is F=MA. At the speed of light, the mass of an object would be infinite, and the force required to accelerate it to that speed would also be infinite. Because of the magnitude of the force, the mass would collapse into a black hole long before reaching light speed. So, it is not possible to achieve mass and acceleration at light speed.
Relativity and light speed present a confusing concept in physics. My students always came into the physics class with prejudice based on what their family had told them. The relatives scared the students into thinking physics class was going to be hard. I always began the year by telling the students that physics was the easiest class they would ever take as long as they learn to speak algebra.
One year, a student enrolled in my physics class who had escaped Viet Nam and spoke virtually no English. The guidance counselor questioned how the student could handle my class with the language handicap. The young man smiled and said through his interpreter, “But I speak excellent algebra!” That was true, and he was my best student that year. Einstein’s theory of relativity is a physics unit that invokes fear in many minds, but it’s easy to understand if you know a little algebra.
The problem is not understanding relativity and light speed. The problem is believing it. Relativity begins with TWO BASIC POSTULATES:
THE FIRST is that the laws of physics are the same no matter where you are or what you are doing. If you are sitting in your chair reading this, all the laws of physics work very well. When you drop an object, it falls in accordance with the laws of motion. If you were in an airplane traveling near the speed of sound and you drop the same object, it would fall the same way as it did when you were sitting still.
THE SECOND postulate tells us that the speed of light is a universal constant. This one is easy to understand, but very hard to believe. Suppose I were in a rocket traveling toward you at half the speed of light. If I turn on my headlights, the light beam will travel at the speed of light. You are sitting still and measuring the speed of the light beam. What would your measurement be? You might be tempted to say, “The speed of the rocket, 0.5 times light speed, must be added to the speed of light. The answer would be 1.5 times light speed.” What Einstein’s postulate says is that you would measure it to be the speed of light–186,000 miles or 300,000,00 meters per second. That’s because the speed of light is a constant, independent of the motion of the light source or the observer.
Light speed is designed to be a universal constant according to Einstein relativity equations. You say, “How can that be?” According to Einstein, time is a created thing that depends upon the motion of the observer. As you go faster, time slows down. The algebraic equation is that the time you experience (t’) equals the time you would experience at rest (t), divided by the square root of 1 minus the velocity (v) squared divided by the speed of light (c) squared. Notice that the velocity cannot be higher than the speed of light. If it were, the denominator would be the square root of a negative number, which is not possible. If you don’t understand the equation, understand that time is not a fixed thing. It changes with velocity. The faster you go, the slower time passes. At light speed, time would stop.
Science fiction writers have suggested that this is a way to build a time machine. That won’t work, because time doesn’t reverse. Since the speed of light is always the same for all observers, time gets slower and slower but never stops. This is not wild speculation. Experiments at very high speeds in particle accelerators have verified what we have briefly sketched here.