2 Fundamentals of General Relativity2.1 Concept Questions1. What assumption of GR makes it possible to introduce a coordinate system?2. Is the speed of light a universal constant in GR? If so, in what se nse?3. What does “locally inertial” mean? How local is local?4. Why is spacetime locally inertial?5. What assumption of GR makes it possible to introduce clocks and rulers?6. Consider two observers at the same point and with the same instantaneous velocity,but one is accelerating and the other is in free-fall. What is the relation between theproper time or proper distance along an infinitesimal interval measured by the twoobservers? What assumption of GR implies this?7. Does the (Strong) Principle of Equivalence imply that two unequal masses will fall atthe same rate in a gravitational field? Explain.8. In what respects is the Strong Principle of Equivalence (gravity is equivalent to ac-celeration) stronger than the Weak Principle of Equivalence (gravitating mass equalsinertial mass)?9. Standing on the surface of the Earth, you hold an object of negative mass in yourhand, and drop it. According to the Principle of Equivalence, does the negative massfall up or down?10. Same as the previous question, but what does Newtonian gravity predict?11. You have a box of negative mass particles, and you remove energy from it. Do theparticles move faster or slower? Does the entropy of the box increase or decrease?Does the pressure exerted by the particles on the walls of the box increase or decrease?12. You shine two light beams along identical directions in a gravitational field. The twolight beams are identical in every way except that they have two different frequencies.Does the Equivalence Principle imply that the interference pattern produced by eachof the beams individually is the same?13. What is a “straight line”, according to the Principle of Equivalence?14. If all objects move on straight lines, how is it that when, standing on the surface ofthe Earth, you throw two objects in the s ame direction but with different velocities,they follow two different trajectories?15. In relativity, what is the generalization of the “shortest distance between two points”?116. What kinds of general coordinate transformations are allowed in GR?17. In GR, what is a scalar? A 4-vector? A tensor? Which of the following is ascalar/vector/tensor/none-of-the-above? (a) a set of coordinates xµ; (b) a coordinateinterval dxµ; (c) proper time τ?18. What does general covariance mean?19. Why is it important to define covariant derivatives that behave like tensors?20. Is covariant differentiation a derivation? That is, is covariant differentiation a linearoperation, and does it obey the Leibniz rule for the derivative of a product?21. What is the covariant derivative of the metric tensor? Explain.22. What does a connection coefficient Γκµνmean physically? Is it a tensor? Why, or whynot?23. An astronaut is in free-fall in orbit around the Earth. Can the astronaut detect thatthere is a gravitational field?24. Can a gravitational field exist in flat space?25. How can you tell whether a given metric is equivalent to the Minkowski metric of flatspace?26. How many degrees of freedom does the metric have? How many of these degrees offreedom can be removed by arbitrary transformations of the spacetime coordinates,and therefore how many physical degrees of freedom are there in spacetime?27. If you insist that the spacetime is spherical, how many physical degrees of freedom arethere in the spacetime?28. If you insist that the spacetime is spatially homogeneous and isotropic (the cosmologicalprinciple), how many physical degrees of freedom are there in the spacetime?29. In GR, you are free to prescribe any spacetime (any metric) you like, including met-rics with wormholes and metrics that connect the future to the past so as to violatecausality. True or false?30. If it is true that in GR you can prescribe any metric you like, then why aren’t youbumping into wormholes and causality violations all the time?31. How much mass does it take to curve space significantly?32. What is the relation between the energy-momentum 4-vector of a particle and theenergy-momentum tensor?233. It is straightforward to go from a prescribed metric to the energy-momentum tensor.True or false?34. It is straightforward to go from a prescribed energy-momentum tensor to the metric.True or false?35. Does the Principle of Equivalence imply Einstein’s equations?36. What do Einstein’s equations mean physically?37. What does the Riemann curvature tensor Rκλµνmean physically? Is it a tensor?38. The Riemann tensor splits into compressive (Ricci) and tidal (Weyl) parts. What dothese parts mean, physically?39. Einstein’s equations imply conservation of energy-momentum, but what does thatmean?40. Do Einstein’s equations describe gravitational waves?41. Do photons (massless particles) gravitate?42. How do different forms of mass-energy gravitate?43. How doe s negative mass gravitate?32.2 What’s important?This section of the notes adopts the traditional coordinate-based approach to GR. Theapproach is neither the most insightful nor the most powerful, but it is the fastest route toconnecting the metric to the energy-momentum content of spacetime.1. Postulates of GR. How do the various postulates imply the mathematical structure ofGR?2. The road from spacetime curvature to energy-momentum:metric gµν→ connection coefficients Γκµν→ Riemann curvature tensor Rκλµν→ Ricci tensor Rκµand scalar R→ Einstein tensor Gκµ= Rκµ−12gκµR→ energy-momentum tensor Tκµ3. 4-velocity and 4-momentum. Geodesic equation.4. Bianchi identities guarantee conservation of energy-momentum.42.3 The postulates of General Relativity1. Spacetime is a 4-dimensional manifold2. The (Strong) Principle of Equivalence3. Einstein’s Equations4. No torsion2.3.1 Spacetime is a 4-dimensional manifoldA 4-dimensional manifold is defined mathematically to be a topological space that is locallyhomomorphic to Euclidean 4-space R4.This postulate implies that it is possible to set up a coordinate system (possibly in patches)xµ≡ {x0, x1, x2, x3} (1)such that each point of (the patch of) spacetime has a unique coordinate.Andrew’s convention:Greek dummy indices label curved spacetime coordinates.Latin dummy indices label locally inertial coordinates.2.3.2 (Strong) Principle of Equivalence (PE)“The laws of physics in a gravitating frame are