http://preposterousuniverse.com/grnotes/ has a lot of good notes on GR and links toother 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:can they think of other examples in physics? F = ma versus Lagrangian or Hamiltonianmechanics; wave versus matrix versus path integral quantum mechanics; quaternions(!)versus vector electromagnetism. In the case of GR, there is the geometric approach, good forinsight and reasoning, and the action approach, probably better for trying to unify gravitywith the other forces. In all these examples, a common theme is that the predictions hadbetter b e the same. Similarly, although, say Newtonian mechanics is based on a completelydifferent set 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 with observables, is the most fundamental point of theories, in my opinion.Therefore, I will present 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 t hat 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 d etail about th e formalismand equations of GR than I did into particle interactions. The reason is that you aren’tnecessarily going to see GR anywhere else, so I’d like th is part of the course to be moreself-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 accelerometerwould measure 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 co ordinatetransformation that will make it look flat everywhere.(5) However, THE most important principle of GR is that in a su fficiently 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 th e 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 whetheryou are at rest in a gravitational field or are being accelerated in flat spacetime. Theequivalence principle means that in practice one of the best ways to do calculations in GRis to do them in the local inertial frame and then use well-defined transformations betweenthe local and the global frame.(6) All forms of energy gravitate. In the Newtonian limit, rest mass is overwhelminglythe dominant component, b ut 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 sur face 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 depend s on the size of t he triangle.Another example is that if you take a vector on a flat plane and transport it parallel toitself, you can move it around the plane to your heart’s content and when you bring it backto the starting point it will have the same orientation it did before. This is not the case ona sphere!Note, however, that (in good analogy to GR!), on a small enough region of a sphere youcan treat it as flat. We need to develop a formalism that can handle curvature like this,except in four dimensions (t hree spatial, one time). This is the formalism of geometry incurved spacetime.Geometric objects.—Let’s start by defining some geometric objects. Bear with me forthe first couple, which seem obvious but lay the groundwork for the less obvious sequels.Event.—First we have an event. An event is effectively a “point” in spacetime. Moregenerally, if you have an N-dimensional space, you need N numbers to label it uniquely.For example, in two dimensions you need two numbers; e.g., x and y for a plane, or θ andφ for the surface of a sphere. For spacetime, you need four numbers: e.g., t, x, y, and z.Naturally, the essence of the event isn’t changed if you relabel the coordinates. I want tostress this, because something that is obvious for events but may not be obvious for someother geometric objects is that although when you finally calculate something you maychoose a coordinate system and br eak things into components, there is also an independentreality (well, within the math at least!) of the objects. Going with the coordinate-freerepresentation has proved very helpful in proving theorems about GR, but when doingastrophysics it is usually best to investigate components in some given system.Vector.—Next, consider a vector. In flat space, this is easy. Using our previousdefinition, we can simply think of a vector as an arrow connecting two events. As longas we define the arrow to be a straight line, there is no ambiguity, regardless of how farseparated the two events are. Again for concreteness, let’s think of two dimensions andCartesian coordinates, so the events are labeled by