But when we look for spacetime curvature, it is very hard to find - it was only after Einstein's theory that it was realized that even very small curvatures are equivalent to enormously strong gravitational fields. It is for this reason that we see gravitational fields around matter - matter itself contains energy, according to the famous relationship E = mc2. What is more, the cause of the curvature turns out to be energy (more precisely, what physicists call 'stress-energy'). One of the key insights in Einstein's extraordinary General Theory of Relativity is that spacetime is indeed curved, and that this curvature has to do with gravity - in fact, in a certain sense, gravity IS spacetime curvature. All we can do is make measurements to see if the spacetime we live in is flat or not. Now again, it is crucial that we cannot tell "just by looking" to see if our spacetime geometry is flat (to do this, we would have to live in 5 dimensions or higher). What now of us? We of course do not live in 2 dimensions - indeed, as understood by Einstein and Minkowski, our world is the 4-dimensional one of spacetime.
But it is not allowed to "look at" these figures from "outside". From the pictures above, we see that by doing simple measurements a 2-dimensional being can figure out if it is living in a curved or a flat geometry. ABOVE RIGHT:Ī triangle in a spherical surface - the angles do not add up to 180 degrees.Īgain, you must resist the temptation of thinking in 3d here - imagine that our being cannot sense anything outside the 2d surface. We see that if we extend them as straight lines, they will eventually meet.ĪBOVE LEFT: What happens if we try to make a square on a sphere, with 4 right angles & equal sides. Two parallel lines on a spherical surface. This is easy - we can look at a spherical surface.
To see what will happen if the surface is NOT flat, let's look at what we would get if now tried to make such measurements on a "curved" surface. The answer is that the only way to do this is to make measurements of geometric objects like those shown above. But if we are 'stuck' inside the surface, how can we check these? It is not as if we can "look down on the triangle from outside" because for us there is no outside. And certainly we expect that the sum of the internal angles of a triangle is 180 degrees. Now objects in a flat space obey things we all take for granted - like, eg., Pythagoras's theorem, or the idea that 2 parallel lines never meet, always staying the same distance apart or that the area of a square of side L is equal to L2. How can we tell, from inside, whether our space is "curved" or not? In fact there is only one way: we have to make measurements - of lengths, angles, and so on - and see whether the results agree with what we would get if the space was "flat". Suppose, however, that we do not live in 3-d space, and that our entire experience is confined to the 2 dimensions of the surface - we don't even know that there exists a world outside the surface, and have no means of doing so. If we look at the surface from outside (ie., from the vantage of our own 3-d space) it is easy to see if this is happening - we could, eg., paint some object on the surface, and see if its shape distorts. Now when a mathematician or physicist talks about "curvature" for some surface, what they really mean is stretching or twisting in a way which changes angles and distances between points in the surface. Suppose we look first at some curved balloon surface - this is a 2-dimensional surface which we see in our own 3-dimensional space. Let's see if we can understand this in a simple way. One of the things that leaps to mind when Einstein is mentioned is the idea of 'curved space', and its association with gravity. Intro The Big Problem Einstein Quantum Curved Spacetime Q.
PITP/ST-JOHNS PUBLIC LECTURES, 2014 -2015 RESOURCE MATERIAL for the EINSTEIN CENTENARY (2015) Centenary of General Relativity: Curved Spacetime
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