The last part of Unit 6 Session 1 was dancing around a really interesting coincidence in our world: There are two ways to measure mass. Even though they are completely different, the values the two ways give just happen to be the same. Coincidence?
All measurements of mass depend on comparing an unknown mass to a standard one. The two ways are
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Comparing masses two ways: gravity and inertia.
The mass of a certain volume of water could be
used as the "standard". |
- Put the object you're measuring on a scale and compare the force of gravity on it to the force on another standard mass.
- Try to accelerate it with a force that you can reproduce and compare the acceleration that you get to what you get when you try to accelerate a standard mass with the same force.
These two measurements always give equivalent results. This is so surprising that it has been checked over and over again to high accuracy. When such coincidences seem to occur in nature it's probable that it's not a coincidence, but there is something behind it. That's what Einstein thought and he managed to figure out a logical connection.
To understand the basic idea of Einstein's theory of gravitation imagine that you had a little laboratory in a box with no windows. You can put this laboratory on earth and do experiments with balls and carts and pendulums....whatever. Then you could put this imaginary lab-in-a-box in a ship and floating far from any gravitational pull and you'd expect the balls not to fall, the pendulums not to swing etc etc, But now fire up the rockets and make the lab accelerate at 9.8 m/s/s and all your experiments would turn out exactly like they would on earth. Einstein took this as a postulate which he called the principle of equivalence.
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Experiments on an accelerating rocket ship would seem the same as if gravity were present.
(Figure from Wikipedia) |
So gravity acts just like the "fictitious" force that you would feel in an accelerating frame of reference. Similarly the forces you feel on a carousel or centrifuge are proportional to mass, just like gravity — and, in fact, you can use a spinning device to measure mass.
Somehow, mass creates something like an acceleration. This is accomplished by having mass cause space to warp. Everything travels in a straight line -- which means the path of shortest distance between two points -- but the space is warped by mass so that the lines are not straight in the Euclidean sense.
To see how this works in 3 dimensions we have to imagine it in 2D where we can visualize warped space. Then the analogy to 3D can be accepted. Formally what is done is that the mathematics of warped space is developed that can be used in any number of dimensions so we won't have to be able to visualize what's going on in order or predict what's happening.
So in 2 dimensions a flat universe would have familiar properties as postulated by Euclid: Parallel lines never meet, the interior angles in a triangle add up to 180° and so forth. Now imagine that the flat space is warped somewhat like a fabric that is poked with your finger.
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Mass causes the space near it to warp.
This is a 2D model to help you imagine what's going on in 3D.
(Figure from Wikipedia.) |
In this case, draw a triangle, add up the angles and the sum is
not 180°. What's important here though is that the shortest distance between two points tends to bend to conform with the curvature of the fabric. The line of shortest distance is called the
geodesic and is familiar on our spherical earth as what is called the great circle route that airplanes take when trying to navigate. In Einstein's universe the curvature of space depends on the mass that is present in the space and the trajectory of an object following the geodesic in the curved space is about the same as what Newton's law of gravitation predicts.
Notice I said "about" the same. Actually there are some small differences between Newton's gravity and Einstein's. One example is the precession of the perihelion of Mercury's orbit. The orbit of Mercury does precess as Einstiein's theory predicts and Newton's does not.
Another prediction is that light should be pulled by gravity. Imagine a little hole in one wall of the lab in the rocket ship. If light comes in that hole while the ship is accelerating then the light will hit the other wall a little lower down because of the ship's acceleration. The light beam will seem to "fall". So by the principle of equivalence, gravity should cause light to "fall". A very careful observation of the light of stars that comes close to the sun shows that this light does bend because the sun's gravity attracts it. This observation has to be done during a total eclipse of the sun and the first time it was done was during an expedition organized by Sir Arthur Eddington in 1910 to test Einstein's prediction. The bending was confirmed in that the stars whose light came near the sun seemed to shift position from where they should have been.
. It tells the story of the project to establish the value of the metre by accurately triangulating the distance from Dunkirk to Paris to Barcelona from hilltop to church steeple to hilltop etc. Two expeditions set out to do it ---- and they did. We still use the value they determined. (This book is a novelization based on the real project.)
, Michio Kaku
, George Gamow
, George Gamow
, Lincoln Barnett (introduction by A. Einstein)
