http://pancake.uchicago.edu/∼carroll/notes/ has a lot of good notes on GR and linksto other pages.General RelativityPhilosophy of general relativity.—As with any major theory in physics, GR has beenframed and derived in many different ways, each giving their own insight. Ask class: canthey think of other examples in physics? F = ma versus Lagrangian or Hamiltonian me-chanics; wave versus matrix versus path integral quantum mechanics; quaternions(!) versusvector electromagnetism. In the case of GR, there is the geometric approach, good for insightand reasoning, and the action approach, probably better for trying to unify gravity with theother forces. In all these examples, a common theme is that the predictions had better bethe same. Similarly, although, say Newtonian mechanics is based on a completely differentset of philosophical principles than quantum mechanics or relativity, in the big, slow-moving,weak-gravity limit the predictions of all those theories are the same. This, the contact withobservables, is the most fundamental point of theories, in my opinion. Therefore, I willpresent things in a way designed for calculation.Another point about general relativity is that it is the least confirmed of our currentfundamental theories. A major reason for this is that its most dramatic effects only showup in extremely strong gravity, such as near black holes and neutron stars. This gives itspecial status, and means that astronomical observations may have the most to contributeto fundamental physical understanding in the realm of strong gravity.Finally, let me say that I plan to go into a little more detail about the formalism andequations of GR than I did into particle interactions. The reason is that you aren’t necessarilygoing to see GR anywhere else, so I’d like this part of the course to be more self-contained.Fundamental GR concepts(1) As in special relativity, space and time are both considered as aspects of spacetime.However, whereas in special relativity spacetime is “flat” (in a sense to be defined later), ingeneral relativity the presence of gravity warps spacetime.(2) The natural motion of objects is to follow the warps in spacetime. “Matter tellsspace how to curve and space tells matter how to move.” An object that is freely falling (i.e.,following spacetime’s warps) does not “feel” force, meaning that an accelerometer wouldmeasure zero. The path of a freely falling particle is called a geodesic.(3) The only “force” in this sense that can be exerted by gravity is tidal force. That is,if an object has finite size, different parts of it want to follow different geodesics, and thesedeviate. Geodesic deviation is the GR equivalent of tidal forces.(4) Because of this deviation, global spacetime is not flat and there is no coordinatetransformation that will make it look flat everywhere.(5) However, THE most important principle of GR is that in a sufficiently small regionof spacetime (small spatial scale, small time interval), the spacetime looks flat. This meansthat there is a local inertial frame that can be defined in that small patch of spacetime. Inthat local inertial frame, all the laws of physics are the same as they are in special relativity(electrodynamic, hydrodynamic, strong+weak nuclear, ...)!! This is called the equivalenceprinciple. There is a classic elevator analogy for this principle, which says that if you are inan elevator and you feel like you are being pushed towards its floor, you can’t tell whether youare at rest in a gravitational field or are being accelerated in flat spacetime (see Figure 1).The equivalence principle means that in practice one of the best ways to do calculationsin GR is to do them in the local inertial frame and then use well-defined transformationsbetween the local and the global frame.(6) All forms of energy gravitate. In the Newtonian limit, rest mass is overwhelminglythe dominant component, but in ultradense matter other forms can be important as well.The Mathematics of Curved SpacetimeConsider a two-dimensional space. We know that there are differences between, e.g., aflat plane and the surface of a sphere. One example of this is that on a plane, the interiorangles of a triangle always add to 180◦, whereas on the surface of a sphere the angles alwaysadd to something larger than 180◦, but the actual value depends on the size of the triangle.Another example is that if you take a vector on a flat plane and transport it parallel to itself,you can move it around the plane to your heart’s content and when you bring it back tothe starting point it will have the same orientation it did before. This is not the case on asphere!Note, however, that (in good analogy to GR!), on a small enough region of a sphereyou can treat it as flat. We need to use a formalism that can handle curvature like this,except in four dimensions (three spatial, one time). This is the formalism of geometry incurved spacetime, and we encountered the basics (scalars, events, vectors, tensors) in oursecond lecture on special relativity. Let us, however, emphasize one particular point aboutfour-vectors in curved space:Vector.—In flat space, vectors are easy. Using our previous definitions, we can simplythink of a vector as an arrow connecting two events. As long as we define the arrow tobe a straight line, there is no ambiguity, regardless of how far separated the two eventsare. Again for concreteness, let’s think of two dimensions and Cartesian coordinates, so theevents are labeled by their x and y coordinates and a vector between them also has an xand y component. Now, think of two points (aka events) on the surface of a sphere. Howdo we now define a vector? Not easy. First, we need to decide what a straight line is. AFig. 1.— The equivalence principle: local measurements can’t distinguish between acceleration andgravity. From http://www.desc.med.vu.nl/Graphics/Equiv principle.gifgood choice is a great circle. Then, however, we have a problem. Every pair of points ona sphere is connected by at least two great circles, and antipodal points are connected byan infinite number of great circles! That shows that in curved spacetime, vectors can’t bedefined for two points that are distant from each other. Therefore, we define vectors only asinfinitesimal quantities. Here again, we’ve defined a vector as a line between two points (nowinfinitesimally close), but a vector has its own existence independent of points or events