In science fiction movies, spaceships often travel between stars in a few days. Traveling between stars so quickly, however, requires traveling at speeds faster than the speed of light, which according to special relativity is not possible. Why is the speed of light the ultimate speed limit in the universe?
Mass Increase and the Speed of Light Limit
As seen by an outside observer, the mass of a moving object increases with its speed. According to the mathematical equations describing this mass increase, the mass becomes finite at the speed of light. Newton's second law of motion says the force required to accelerate an object is proportional to its mass. It therefore takes an infinite force to accelerate an infinite mass. An infinite outside force would be needed to accelerate an object to the speed of light.
Because there can be no infinite forces, no object with mass can reach the speed of light. Photons of light can travel at the speed of light because they have zero mass.
Lorentz Contraction and the Speed of Light
Lorentz contraction predicts that a moving object will appear shorter in the direction of its motion, as seen by an outside observer. Mathematically an outside observer sees an object moving at the speed of light as having a zero length.
Time Dilation and the Speed of Light
In special relativity time passes more slowly for a moving object as seen by an outside observer. In the mathematical description of time dilation an outside observer would see time stop for an object moving at the speed of light.
Relativity theory does not allow anything to move faster than light. If it were, however, possible for an object to move faster than light, then an outside observer would see time moving backwards for that object. These time dilation effects lead to the well known limerick describing relativity:
- There was a young lady named bright
- Whose speed was much faster than light
- She set out one day
- In a relative way
- And returned on the previous night
Mathematical Details
The mathematical formulas for special relativity use the factor gamma, defined by:
gamma = 1/(√(1-v²/c²))
where, v is the speed of the moving object, c is the speed of light, and √ represents the square root of the quantity in parenthesis.
For an object with a speed equal to the speed of light, v divided by c becomes 1. The quantity in the square root is zero. Mathematically dividing by zero is not allowed, but dividing by a very small number gives a very large number. Therefore in the limit as v gets close to c, the square root gets close to zero and gamma gets close to infinity. In this way, mathematically, when v=c, gamma =infinity.
To find the mass of a moving object, its rest mass is multiplied by gamma. Hence the mass of an object moving at the speed of light is infinite.
Lorentz contraction requires dividing the rest length by gamma. Hence at the speed of light a moving object appears to have a zero length as seen by an outside observer.
A time interval is multiplied by gamma to find the time interval as seen by an outside observer. When v=c, time intervals seen by outside observers become zero. Hence time appears to stop, as seen by outside observers.
In a strict mathematical sense the gamma factor only forbids objects from moving at the speed of light. Moving faster than light would however require going through at least an instant traveling at the speed of light, violating special relativity. Mathematically when v exceeds c, gamma becomes what mathematicians call an imaginary number. This mathematics opens up the very speculative possibility of objects that only travel faster than light and cannot travel more slowly than light. Physicists call such particles tachyons. There is not one shred of experimental evidence that tachyons actually exist.
The fact that gamma becomes infinite at the speed of light forbids objects having mass from moving at the speed of light. Hence the speed of light is the ultimate speed limit in the universe.
Further Reading
Einstein, A., "On the Electrodynamics of Moving Bodies" Annalen der Physik, 1905, Translated and reprinted in The Principle of Relativity, Dover, 1952.
Tipler, P.A., Modern Physics, Worth, 1978.
Serway, R.A., Moses, C.J., and Moyer, C.A., Modern Physics, Thomson, 2005.
Einstein's Special Relativity and Mass Energy Equivalency, E=mc^2
Paul A. Heckert - I have a Ph.D. in astrophysics, over 30 years experience teaching physics and astronomy, and over 60 published research articles.
