Albert Einstein published his special theory of relativity in 1905. Einstein based this theory on the assumption that the speed of light in a vacuum is a fundamental constant that has the same value for all observers in reference frames that are not accelerating.
Special relativity theory predicts several unexpected effects which include:
- mass increase with speed
- Lorentz contraction
- time dilation
- nothing travels faster than the speed of light
- mass energy equivalency, E=mc²
Mass Increase with Speed
Imagine Einstein standing at rest on a scale that measures his mass as 100 kilograms. Now imagine Einstein running wearing special superhero grade rocket powered running shoes that allow him to run at nearly the speed of light. As Einstein runs across the scale, it will measure his mass as being more than his 100 kilogram rest mass. Exactly how much more depends on Einstein's speed.
The scale is not broken. The increased scale reading is a consequence of Einstein's special theory of relativity.
The mass of an object when it is at rest, or more properly when measured in a reference frame that is moving at the same velocity as the object, is called the rest mass. When the mass of the same object is measured in a different reference frame, in which the object is moving, the object's mass will be greater than its rest mass. So the scale measures a greater mass when Einstein runs across it that when he stands on it.
This mass increase is not measurable at speeds that humans can usually run. The amount of mass increase depends on the speed but does not become significant until an object is moving at about 30,000 kilometers per second, which is 10% of the speed of light.
Why Mass Increases with Speed
A thought experiment, like those Einstein used, can help illustrate why an object's mass must increase as its speed increases. Imagine two boys playing catch in a moving rocket and an observer at rest watching the rocket zoom past. The boys each throw a ball and the balls collide in midair. According to the law of conservation of momentum the total momentum of the two balls must be the same (conserved) before and after the collision. Momentum is the mass multiplied by the velocity, and velocity is distance over time.
The collision occurs inside the moving rocket, and the boys inside the rocket see that the momentum of the colliding balls is conserved. However the observer at rest outside of the rocket sees different distances the balls travel and different times because of the Lorentz contraction and time dilation. The observer at rest therefore measures different velocities for the balls. If the observer at rest measures the same masses for the balls, then momentum (mass times velocity) would not be conserved in the rest reference frame. If the observer in the rest reference frame measures different masses for the balls, however, the law of conservation of momentum can be followed in both reference frames.
Einstein therefore concluded that the mass of moving objects must increase as seen by an observer at rest, but not as seen by an observer moving with the objects.
Mathematical Formula for Mass Increase
In special relativity theory, the mathematical formulas often use the factor gamma, which is 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.
The relativistic mass of a moving object is the rest mass multiplied by gamma. Mathematically when v=c, gamma becomes infinite. Hence an object's mass would be infinite if it were moving at the speed of light.
The relativistic mass increase with speed has implications for the relativistic momentum and energy, which relate to Einstein's famous equation, E=mc².
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.
Paul A. Heckert - I have a Ph.D. in astrophysics, over 30 years experience teaching physics and astronomy, and over 60 published research articles.
