1. General Relativity Physics Honours 2006 A/Prof. Geraint F. Lewis Rm 557, A29 gfl@physics.usyd.edu.au Lecture Notes 10

2. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 10 Stress-Energy Tensor Chapter 22 Unlike classical gravity, (virtually) all sources of energy produce gravitational effects through the warping of spacetime. The distribution of energy is described by the stress-energy (or energy-momentum) tensor. To understand the tensor, consider a box at rest with respect to you. The number density of particles in the box is simply What if the box was moving relative to you (assuming SR)?

3. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 10 Densities We can define a number-current 4-vector The spatial components correspond to number current density, and can be used to calculate the flow of particles per unit time across an areas through The conservation of particle number within an elemental volume can be simply written as

4. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 10 Energy & Momentum A 3-d volume in 4-d spacetime possesses a normal vector (see section 7) and the volume is given by The number of particles in the volume is given by When considering the flow of energy and momentum through the volume, then we need a similar expression, namely Where T  is the stress-energy tensor. To understand what this means, consider flat spacetime at a constant time; this is a 3- d space with n  =(1,0,0,0).

5. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 10 Energy-Momentum Tensor Assuming this we get Let’s look at our particles in a box again And we can guess that the general form for this is The non-t terms are shearing terms.

6. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 10 Energy-Momentum As with our particles in a box, we can derive an conservation equation for our energy momentum tensor when considering flat spacetime This is actually 4 equations! The time component gives The remaining three give the pressureless Navier-Stokes eqn.

7. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 10 Perfect Fluids If we are at rest with respect to a perfect fluid then In flat spacetime, we can extend this to a moving fluid so It should be clear that in the rest frame, this becomes the above rest frame expression. How do we generalize to curved spacetime? Generally,  -> g and derivatives go to covariant derivatives. The problem is that our conservative laws are only local (and we lose our global concepts of conserved quantities).

8. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 10 Einstein Equation What is the simplest equation we can build from geometry (Riemann tensor, Ricci tensor, Ricci scalar) and our stress energy tensor? We can use the Bianchi identities to give And the Newtonian limit gives us , and we (finally) get the Einstein equation