GRAVITY AND GENERAL RELATIVITY by Richard Taylor RASC Ottawa Centre and PAS Member
I teach high school Physics, and sadly, most of what I teach is three or four hundred years out of date. Sure, it's still a useful approximation of the real universe, but all the interesting and more accurate theories that REAL physicists are using are generally mentioned only as an aside or a briefly covered topic at the end of the term. Most of the course content is Galilean and Newtonian physics. Part of the reason for this is that the mathematics needed for quantum wave equations and general relativity are just too advanced for high school students. One of my long-term goals is to find ways of explaining gravity according to Einstein at a level that high school students can understand and calculate.
Newton's Law of Universal Gravitation is fairly easy to explain and understand: all masses create a gravitational force that attract other masses; the force is proportional to the product of the masses and inversely proportional to the distance between their centers, squared. F = GMm/r2. The capital G is the proportionality constant and is VERY small: 6.67x10-11 in metric units. This means that the force of gravity is only noticeable if at least one of the masses is very large (capital M in the equation), like the mass of the Earth. When both masses are fairly small, the force of gravity is extremely small. The force of gravity between you and your friend, 1 meter apart, is only about 0.00000033 Newtons.
Newton had some other famous theories in addition to his Law of Universal Gravitation; you've probably heard of Newton's Laws of Motion. The First Law is inertia: objects tend to stay at rest or at a constant velocity (speed and direction). The Second Law is about forces: a = F/m. When a force is applied to an object, it accelerates, but the greater the mass, the less acceleration. Because larger masses tend to keep on doing what they were doing and "resist" the effects of external forces, physicists call it "inertial mass." (The Third Law is about action and reaction, but I'm not going to use that in this article.) Consider the Second Law of Motion and Newton's Law of Universal Gravitation. You've probably never been bothered by the fact that "m" for mass occurs in both of these equations. Mass resists the action of forces yet causes a force of gravitation. Einstein thought this was strange. Why should "inertial mass" be exactly the same as "gravitational mass"? What has gravity got to do with inertia? Maybe gravity isn't a force at all; maybe it's just our resistance to being accelerated. Einstein came up with a principle of equivalence: he made the claim that for an observer in a closed, windowless room, there would be no way of telling the difference between that room sitting on the surface of the Earth and that room accelerating upwards at 9.8 m/s2. Then he combined this idea with some of his recently developed ideas about motion at speeds near the speed of light and made some outlandish predictions. Even though light has no mass, its path would be bent downwards by gravity because light also resists being accelerated. Clocks in a gravitational force field will run slower than clocks not in a gravitational force field because our concept of time is closely linked to our concept of motion. Light escaping from a gravitational force field will have longer wavelengths. Each of these predictions have been tested experimentally and have been supported by observations. Einstein was right: there is no difference between gravity and acceleration.
But Einstein didn't stop there. He then went on to explain WHY gravity is the same as acceleration. That's where things get really complicated. First of all, you need to understand that space and time are closely related in a four-dimensional space-time. Then you need to understand that this four-dimensional space-time doesn't always operate geometrically according to the rules of geometry you learned in school. When there is mass and energy present, the geometry is curved.
Let's see if I can explain each of these ideas. We normally think that we live in a three-dimensional world. The position of any object can be described with three measurements from an observer or from a standard origin. The top of the door is 4 meters ahead of me, 1 meter to my left and 0.2 m above my head. However, cats could tell you that there are actually FOUR measurements of importance: the bed is 1 m ahead of the food dish, 3.5 m to the left, 4 m above and belongs to Skittles at 9:00 am (while the sun is on it). Skittles will fight off any competition at this particular location in space-time, but makes no objection to Cleo using the same spatial location at 10:00 pm (a different space-time coordinate.) Skittles' preferred space-time coordinate then shifts to 4 m ahead of the food dish, 1 m right, 0.5 m up and 10:00 pm. Sometimes it helps to draw or think of a graph of space-time. Most humans live fairly two dimensional lives (we rarely fly up into the third spatial dimension.) So you could imagine time as being the third dimension. We are all moving along in time at a constant speed of one second per second. Your path in the space-time graph moves around from place to place on the two-dimensional surface while continually moving "upwards" in the time dimension. Your life is an 80- or 90-year long "worm" in space-time, starting as a tiny fertilized egg at the space-time event of your conception, getting bigger and wriggling around through space-time until you die, at which event your body stops moving through space and gradually peters out through time.
When Einstein put together his ideas about Special Relativity (the consequences of the speed of light being the same for all observers), he found that observers who are moving at different speeds through space will not agree about the size, location or even the timing of events in space-time. What happens is that the axes of the space-time graph for the different observers get slanted with respect to each other. Part of this is obvious: I'm sitting on my sofa at rest in space. My path in space-time is a straight line parallel to the time axis. Kelly is flying over my house. She is sitting in her seat feeling quite at rest also. But from my point of view, Kelly's path in space-time is slanted - she moves 150 m horizontally for each second she moves in time. The amount of slant obviously depends on her speed, but not-so-obviously also depends on our choice of how we put the numbers on the axes. To show the Einsteinian physics, the best choice is to make each second of time equal to one light-second of spatial distance - 300,000 km! But if we do that, Kelly's 150 meters per second is hardly slanted at all! That's right. Special Relativity is not noticeable at normal human speeds.
Now let's think about a more complicated motion: I take my Physics students to Canada's Wonderland and make them observe the Physics of the rides there. One of them is a large rotating wheel. You stand along the circumference of the wheel and when it spins, you feel a force pushing you outwards. We Physics teachers are fond of calling this the "imaginary" centrifugal force because it's really your inertia trying to make you follow a straight line path while the wheel forces you to follow a curved path instead. If you were to draw a space-time graph of someone on this ride, it would look like a corkscrew, moving round and round in two dimensions of space while proceeding as usual at one second per second in time. Just like the preceding example, if we number the axes so that one second is the same size as one light-second, then the curvature of this corkscrew is extremely small, yet the students on the ride feel a considerable centrifugal force.
Finally, we come back to gravity. Einstein explained that mass doesn't create a gravitational force, it just bends space-time a bit. Remember, he predicted correctly that clocks in a gravitational field will run slower? Well, actually he said that the presence of nearby mass (or energy) will change the rate of time. You won't be moving along the time axis at one second per second, but at some slightly slower rate. Another way of thinking about this is to bend the time axis slightly towards the mass. The other axes will also get bent a little, too so that in this bent space-time a "straight" line won't be straight anymore. In fact, the natural path of an object (Newton's First Law) through this bent space-time near a large mass is a slightly curved corkscrew shape. An object following this corkscrew path will feel no force at all (you will feel weightless.) This is the case for the astronauts in the international space station. However, for us poor Earth-bound physicists, the ground forces us to follow a real straight line path through space-time. We are continually being pushed away from our natural space-time path, which would be bending in towards the center of the Earth. THAT is the real force that we feel, not a force of gravity at all, but a force of the floor pushing against our feet and forcing us away from our natural free-fall. Just like the centrifugal force of the students on the whirling wheel, the force of gravity is an imaginary force caused by our inertial tendency to try to follow one path while something else is forcing us to follow a different path.
So that is the General Relativity way of explaining gravity. I hope you have been able to follow the ideas. Unfortunately, (and I find this very frustrating) I cannot write an equation as simple as Newton's to describe the corkscrew shaped space-time path of objects near a large mass. Whenever I have tried to track down the equations, I get bogged down in four-dimensional tensors and matrices that were not my strong point in university math. I'll keep looking, and perhaps I'll be able to find some simplifying approximations or special cases that will allow me to put together a high school level formula. There are some neat new ideas about General Relativity and the implications of curved space-time. Take a look at the August 2009 issue of Scientific American. You'll find a very well written article about "swimming in spacetime" (Adventures in Curved Spacetime by Eduardo Gueron.) Using some analogies to the curved surface of the Earth, the author explains how it is possible to move through curved space without the need for any rocket or external force, but simply by changing shape - moving your arms and legs around in a swimming motion. No practical applications yet, but who knows? Perhaps someday we'll have spaceships that wriggle and swim their way from planet to planet without any need for wasteful rocket fuel.