
Facts of the Matter
Richard Brill

Einstein's theories, for nongeniuses
EINSTEIN'S General Theory of Relativity is expressed in mathematical terms that are beyond most of us. But basic understanding of its fundamentals does not require mathematics at all. It merely requires understanding the unexpected relationship between gravity and inertia as expressed by the principle of equivalence.
If you've ever felt heavier or lighter in an elevator as it starts and stops, then you have experienced the principle of equivalence. You not only feel heavier or lighter, you actually are, and it would show on a scale if you were standing on it in the elevator.
Your body's mass has not changed, and gravity has not become stronger. You are feeling the effect of inertia combined with the effect of gravity.
But what if the elevator was in freefall, or if you were dropped from an airplane inside a box with no windows. What would you feel?
In the early 17th century, Galileo verified through a series of ingenious experiments that in the absence of air resistance, all matter accelerates downward at the same rate regardless of its size or mass. This is now called Galileo's law of freefall.
He also discovered the principle of inertia: in the absence of other influences matter in motion will continue to move in a straight line at a constant speed.
Fifty years later, Newton put the two ideas together in his laws of motion, describing the relationship between force, inertia, and acceleration: Force causes matter to accelerate, while it resists acceleration by virtue of its inertia. He defined "mass" as the measure of inertia.
Newton also used the concept of mass in a gravitational sense in the law of universal gravitation: any two objects exert mutual attractive forces on one another, the amount of force depending on the mass of both objects.
He showed that the same gravity that holds the planets in orbit is also the force that causes objects to fall downward and exert the force we call weight.
The laws of motion and gravitation use "mass" in two different contexts.
On one hand, mass is a measure of inertia. On the other hand, mass attracts other mass through the mysterious force of gravity.
Newton assumed, with no attempt at justification, that inertial mass and gravitational mass were mathematically equivalent. Based on that assumption, he then showed with simple proportions how the equivalence explained Galileo's observation that all objects fall at the same rate under the influence of gravity regardless of their mass.
An object with twice the mass of another has twice the weight (gravitational force of the earth) trying to accelerate it, but it also has twice the inertia (its mass), which resists being accelerated. The two effects cancel out so that all objects accelerate in freefall at a constant rate.
Physicists wondered about this relationship between inertia and gravity.
What is the connection? Are the two types of mass really equivalent or are they almost equivalent with the difference undetectable within the margin of error of measurements?
Any difference in the rate of freefall of two objects of different mass or composition, however tiny, would mean that inertial mass and gravitational mass are not equivalent, and would present an awkward situation since much of physics relies on that equivalence.
Ordinary pendulums were the most precise way to measure freefall in Newton's time and for two centuries thereafter. The motion of a pendulum depends on the constant acceleration of gravity, but can test the principle of equivalence to no better than about one part in 100,000.
Many ingenious experiments of various types were conducted over the years with increasing precision. By the beginning of the 20th century, the equivalence had been confirmed to one part in one billion. Since then, with advancements of measurement technology, tests of the principle with different experimental techniques, and with different substances, have verified the equivalence to one part in one trillion.
Although the equivalence of inertial and gravitational mass may seem like a trivial concept, it has much broader and deeper meaning.
In everyday life, the equivalence does not matter at all, but its deeper implications were recognized by none other than Albert Einstein as he pondered his Special Theory of Relativity, which was published in his "miracle year" of 1905.
One day in 1907, while sitting in his office in Bern, Switzerland, he realized in yet another brilliant flash of insight that if he were in freefall (think of a freefalling elevator if its cable breaks), he would be unable to feel his own weight. He later recalled that this was the "happiest moment in his life," for he understood that this simple idea was the key to extending the Special Theory of Relativity to include the effect of gravitation and accelerated motion.
It made a connection between light and the presence of matter, and ultimately led to Einstein's General Theory of Relativity. The General Theory redefined gravity as due spacetime that is warped by the presence of matter, and only appears to move in curved paths due to our inability to detect the spacetime distortions with our five senses.
Einstein took as a supposition that the gravitational mass and inertial mass were truly equivalent, and then performed what he called a "thought experiment" to see what consequences might follow.
He soon realized that the nature of any constant acceleration was such that no experiment could distinguish between weight in a stationary reference frame in a gravitational field and the inertial response to acceleration (such as in an elevator being pulled at a uniformly accelerating rate).
"Down" in the accelerated reference frame would be in the opposite direction to the acceleration, just as we feel pushed backward in the car seat when the car accelerates forward.
Then in another flash of insight Einstein realized that to an observer in an accelerating reference frame any object in motion would also appear to experience a force in the "downward" direction. This would be not only be true of material objects, but also of light.
To illustrate, a ray of light entering through a small hole in the side of the accelerating elevator, although traveling in a straight line in a stationary reference frame, would appear to curve downward in a parabolic arc to an observer in the accelerating reference frame. Because of the high speed of light, (186,000 miles per second, or 670 million miles per hour), at low rates of acceleration this would not noticed, but at high acceleration the light would be notably bent.
Being unable to distinguish between a gravitational field and an accelerated reference frame led to another revolutionary conclusion: light ought to be bent by gravity
This sparked Einstein's decadelong quest to describe gravity mathematically in relativistic terms. He used the principle of equivalence as an axiom to get to general relativity in the same the way that he used the principle of relativity and the constant speed of light axiomatically to develop the Special Theory.
His predictions were verified in 1919 when starlight passing near the sun was deflected as if passing through a lens, just as Einstein's equations had predicted.
Einstein's genius had linked matter, energy, space, time and gravity with the electromagnetic phenomenon that we know superficially as light.
Richard Brill, professor of science at Honolulu Community College (
home.honolulu. hawaii.edu/~rickb), teaches earth and physical science and investigates life and the universe. His column is published on the first and third Sunday of every month. Email questions and comments to
rickb@hcc.hawaii.edu
Richard Brill picks up
where your high school science teacher left off. He is a professor of science
at Honolulu Community College, where he teaches earth and physical
science and investigates life and the universe.
He can be contacted by email at
rickb@hcc.hawaii.edu