David Nidever 11/16/98 Philosophy 350 Dr. Harrison The Concept of Space in Einstein’s General Theory of Relativity In Newton’s physics real and independent existence is ascribed to space, time and matter. Space is thought of as being “at rest” and matter moves at a constant velocity or accelerates with respect to it. In Newton’s laws the space and time coordinates of a particle are crucial in determining the mechanics of the particle. Newton’s first law says that “a particle with no unbalance force acting on it moves with constant velocity”. That is it moves with constant velocity to the space. The second law shows how a particle accelerates, with respect to the space, when acted on by an unbalanced force. For Newton space is independent of matter and can exist on its own. So empty space is possible. Most philosophers and scientists (pretty much the same thing at the time) at the time of Newton didn’t like the idea of assigning physical reality to space, especially empty space. Descartes was one of the most opposed to this idea. His view was that one ought not to ascribe reality to something, like space, which one cannot directly experience. But at the time it was necessary to give mechanics a clear meaning. Normally our conception of space is related to confined space, as in a box. We can stick a certain amount of substances in the box and then no more, but if we put less in there will be some “empty space” left over. So our concept of space has a psychological basis. But what happens when we take away the walls of the box? Is the space still there? To take this example further, let’s imagine that a box within a larger box. Both of them contain a certain/finite amount of space, and there is some space they both contain (the space within the smaller box that is). So far so good. But what happens when the boxes are in relative motion with respect to each other? If the smaller box is moving with respect to the larger box and we were to look at just the small box we would think that it still encloses the same space, but only a variable amount of the space enclosed by the larger box. So each box gets its own space, and they move with respect to each other. With this principle we can imagine having many more boxes and their spaces moving with respect to each other. At the extreme there exists an infinite number of spaces moving with respect to each other. This is nothing like our previous concept of space where this is only one space which is “at rest”. This is not the only objection to ascribing physical reality. As already mentioned by Descartes there is no way to observe space. We can observe extension and distances between objects, but there is no way to observe the absolute and unchanging space coordinates that Newton believed existed. So the idea of a “substantival space” has some important objections to it. The other view of space, a “relational space”, is that space acquires it’s meaning only through objects. When we see material things in the world we can talk about extension and distances, but without them these concept become meaningless. This view was held by Descartes, but his argument for it was weak since he defined space as identical to extension and we give extension meaning through our experiences with bodies. Since we have no experience of empty space we can’t give the concept of extension and therefore space any meaning. Here Descartes enlarges the concept of experience and disregards the possibility of extension without experience. But the relational view of space has it’s own problems. If space acquires it’s meaning through objects that are separated by a certain distance, then how did they acquire this separation in the first place. It seems easy to explain space in an empirical by pointing to experiences of objects, but it doesn’t seem to explain the objects and their “spatial” extensions. Explaining space in this way seems to be circular. And what if we go out to the very last object in the universe and stick out our arm or shoot an arrow? Does this create space? What is the arrow moving in or through, and how come the distance between me and the object is preserved and not gobbled up? If the relational view is the right view then it has some explaining to do. To give the relational view a better footing we need to turn to geometry and Euclid’s axioms. The ancient Greeks and Egyptians knew about geometry and the principles associated with it for a long time. Euclid derived all of these principles from a few basic axioms. The derivations used no other principles or rules, only the ones Euclid started out with. The inferences were very reliable and the axioms seemed undoubtedly self-evident. Mathematicians were satisfied with this for a long time, but many centuries down the road they started worrying about these axioms and epistemological formulations for their truthfulness. The question was whether the axiomatic system could be reduced to simpler and more self-evident statements. One axiom stands out at this point. The axiom of the parallels (which says that there exists only one parallel to a given line through one point) was investigated to see if it could be deduced from another axiom. But this was never successful although many excellent mathematicians worked on the problem. Instead another approach was taken, assume the opposite and deduce a contradiction. The statement that there exist multiple parallels to a given line through one point was introduced and surprisingly no contradiction was arrived at. This was curious. Now there were two axiomatic systems that didn’t entail any contradictions, but seemed opposed to each other. How was this possible. Another astonishment was soon to follow. The statement that there exists no parallel to a line through one point was introduced and no contradictions was arrived at. Now there were three separate axiomatic systems, all opposed to each other, but consistent within themselves. No contradictions or inconsistencies had been arrived at so far, but a proof was needed. And Klein was the one who succeeded in doing the proof. He created non-Euclidian (the other axiomatic systems) counterparts to all concepts of Euclidian geometry, such as points, straight lines, and planes. Now if there were any inconsistencies in the non-Euclidian space there would have to be inconsistencies in Euclidian space. But Euclidian geometry was deducible from very reliable axioms. Therefore non-Euclidian geometries also became viable conceptions of space. What was learned by this was that geometry was really a set of conditional statements. This implicational character of mathematical geometry meant that there were many possible configurations of space. Although there were attempts to show that our space is Euclidian, such as Kant’s visual a priori argument, none were successful. So what kind of space do we have? It was not for mathematical geometry to say. It was for physics to show what our physical geometry is like. Riemann showed that a two dimensional (2D) non-Euclidian space could be seen as the surface of a sphere in a three dimensional (3D) Euclidian space. He also introduced the idea of curvature of space. Space with positive curvature is like a sphere and no parallels to a straight line exist; space with zero curvature is like a plane and only one parallel to a straight line exists; and space with negative curvature is like a hyperbola and many parallels to a straight line exist. But how is it possible to discern which space you are in or what the curvature of your space is? Riemann showed a method of how this could be achieved. It only works if one is in the space, and it is best demonstrated by the sphere. In non-Euclidian spaces the ratio of circumference/radius of a circle is not 2p, and the interior angles of a triangle are not 180 . On the surface of a sphere a triangle can be created by dropping two lines from the north pole, at right angles to each other, to the equator, and then connecting them by a line along the equator. The interior angles of this triangle are 270 , not 180 . With this method physics can discern what axiomatic system of space we have in our universe. But before we get to what type of space we have we must consider a certain situation. Imagine that we have two spaces, G and E. G has a bump in it like in the picture. If we are in the G space we can discover the bump by using Riemann’s method, and they would also discover that surrounding the bump the space is flat like a plane. Now, if shadows were cast of equally placed rods, like the dotted lines, from G to E then people in the E space would be able to tell that the shadows in the middle suffer deformation, by using their own measuring rods. But what if there was a strange force that shrunk the measuring rods just enough so that they would be equal in length to the shadows. Not only the measuring rods, but all objects would suffer this deformation. What kind of measurements would be obtained? Around the bump everything would be the same, but underneath the bump the E people would get the same results as the G people. But what about this strange force. If this force acted differently on different objects then we could tell that this force was there, but if it acts the same way on every object them it’s undetectable. Such a force is called a universal force. At this point the non-Euclidian space and the Euclidian space with a universal force are indistinguishable. At this point we have to introduce a certain type of definition, the coordinative definition. In physical concepts are not just defined by other concepts, but are also coordinated to real objects. The concepts become connected to a real thing and therefore become testable. A coordinative definition is simple stating this concept is coordinated with this particular thing. With coordinative definitions and empirical observations we can test our concepts and geometry with the real world. With coordinative definitions our axiomatic system of geometry becomes a statement about a relation between the universe and rigid rods (measuring rods). As we mentioned earlier we can relate a non-Euclidian space with a Euclidian space with a universal force in it. A popular universal force in physics is gravity. Newton was the first one to accurately describe the effects of gravity on material objects with his equation of universal gravitation. And of course he believed that it was in an Euclidian space that this occurred. So now we have a Euclidian space with a universal field, which we just said was related to a non-Euclidian space. What would that be like? Would such a formulation work? Could it exist? Einstein was the one to try and show that it did work and was actually the correct description of gravity and our space. The guiding principle of Einstein’s General Theory of Relativity is the principle of covariance. It says that an inertial frame in a gravitational field is equivalent to an accelerating field. The reason is the following: Imagine that you are in a small room with no windows. If this booth is suspended by a rope from a building the person you will feel pressure on your feet and legs because gravity is pulling you, and everything else in the room, downwards. But what if the room was being pulled, at a constant acceleration (equal to the acceleration of gravity), by a rocket with the same rope? What would you feel? Because of inertia it takes an unbalanced force to accelerate you and anything else in the room. Therefore the bottom of the room will be pressing on your feet, applying a constant force to you, with the effects as gravity did. So, if you are in a closed room as an observer you can’t tell the difference. This is the principle of covariance. It goes further though. What happens to light when it is observed from an accelerating frame? Light travels at a constant speed in a straight line. If we are accelerating perpendicular to the velocity of the light then it’s motion in the direction that we are moving in will appear to be a curve (because we don’t see ourselves as moving). According to the principle of covariance this should also happen in an inertial frame in a gravitational field. But why would light follow a curved path? And isn’t the speed of light supposed to be the highest velocity possible? Motion along a straight line between two points would be faster than an arc that intersects both points. How can this be resolved? Since we have already seen that a Euclidean space in a universal field such as gravity can be thought of as a non-Euclidian space, then why not try and see if the curved path of light makes more sense in a non-Euclidian space. A Euclidian space can be made into a non-Euclidian space by making transformations in the regions where the space is to be curved. Newton’s equations for the motion of a mass particle in a gravitational field are given by, ü = g (where u is any dimension) If we introduce a transformation u = u’ + (g/2)t2 then, ü = ü’ + g and ü’ = 0 Basically, the motion of a mass particle is transformation from acceleration to constant velocity. This explains the curving of light very well. In a Euclidian space changing velocity or following a curve is accelerating. But with this new transformation we can say that light doesn’t accelerate when it’s motion is a curve, it’s following a path of through the non-Euclidian space at constant velocity. Einstein’s General Theory of Relativity is a lot more complicated than I have summarized here, but the basic idea is that geometry does not change physics, physics changes geometry. That is a large massive object distorts space. This is hard to accept, but as we saw earlier there are many different possible configurations of space, and we can’t tell by math which one it is. Through coordinative definitions and empirical observations we can see if Einstein was right. And so far there has been large amounts of evidence in favor of the General Theory of Relativity. Curving of light, time dilation (which I didn’t explain), the orbit of Mercury, and other observation are in agreement , to a high degree of accuracy, with General Relativity. So what does this say about our earlier questions about the substantival and relational view of space? General Relativity shows that material objects affect space, and not the other way around. This supports the relational view. But what about the space between objects, which is relatively flat? Does it’s existence depend on the objects around it? And what about space itself? As we said earlier there can be many different configurations of space. Doesn’t this mean that there is actually something out there to be configured? And how about our three spatial dimensions? We are bound to these, we can’t break out and stick our hand into a fourth dimension. So it does seem that parts of both views are correct. There is something real, restrictive and configured about space, but material objects also change space and how it is configured. Einstein himself said about the matter: “There is no such thing as an empty space, i.e. a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field.” (By field here he means a generalized field as a field of force such as a gravitational field) Even here, though, there seems to be something substantival, the field. It seems to take on the role of space in Newton’s view, with some changes. In conclusion, space seems to be best explained empirically by reduction to relations of objects, as demonstrated by Einstein’s General Theory of Relativity, but there still remain questions about why space is the way is it, and if its properties (or apparent properties) can be satisfactorily reduced to material objects.