General Relativity. Einstein s Theory of Gravitation. March R. H. Gowdy (VCU) General Relativity 03/06 1 / 26

Transcription

1 General Relativity Einstein s Theory of Gravitation Robert H. Gowdy Virginia Commonwealth University March 2007 R. H. Gowdy (VCU) General Relativity 03/06 1 / 26

2 What is General Relativity? General Relativity is the currently accepted theory of gravity. It has passed every experimental test so far. It is the low energy limit of most proposed theories of everything. General Relativity is the currently accepted theory of space and time. It includes Special Relativity as a limiting case. It provides a detailed theory of measurement. It predicts black holes and gravitational waves. It provides a framework for describing the history of the universe. It does not play well with others (quantum theory in particular). General Relativity is best expressed in the language of di erential geometry. R. H. Gowdy (VCU) General Relativity 03/06 2 / 26

3 What is General Relativity? General Relativity is the currently accepted theory of gravity. It has passed every experimental test so far. It is the low energy limit of most proposed theories of everything. General Relativity is the currently accepted theory of space and time. It includes Special Relativity as a limiting case. It provides a detailed theory of measurement. It predicts black holes and gravitational waves. It provides a framework for describing the history of the universe. It does not play well with others (quantum theory in particular). General Relativity is best expressed in the language of di erential geometry. But we will make do without that. R. H. Gowdy (VCU) General Relativity 03/06 2 / 26

4 The Plan A Reminder of Special Relativity R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

5 The Plan A Reminder of Special Relativity The Equivalence Principal R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

6 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

7 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

8 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

9 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

10 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

11 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. Describing Spacetime R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

12 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. Describing Spacetime Moving reference frames and the metric tensor R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

13 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. Describing Spacetime Moving reference frames and the metric tensor Solving the Field Equations R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

14 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. Describing Spacetime Moving reference frames and the metric tensor Solving the Field Equations The spacetime around a spherical object R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

15 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. Describing Spacetime Moving reference frames and the metric tensor Solving the Field Equations The spacetime around a spherical object The large-scale geometry of the universe R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

16 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. Describing Spacetime Moving reference frames and the metric tensor Solving the Field Equations The spacetime around a spherical object The large-scale geometry of the universe Ripples in spacetime R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

17 The Plan A Reminder of Special Relativity The Equivalence Principal Curved Spacetime Why rubber-sheet pictures are misleading Einstein s Field Equations In vacuum: Directly from Newton s inverse square law of gravity. With matter: Pressure attracts. Describing Spacetime Moving reference frames and the metric tensor Solving the Field Equations The spacetime around a spherical object The large-scale geometry of the universe Ripples in spacetime Testing the predictions of General Relativity R. H. Gowdy (VCU) General Relativity 03/06 3 / 26

18 Spacetime We live in a four dimensional geometry. Three space dimensions and one time dimension. What we think of as an object is represented by its history or worldline. What we think of as the speed of an object is the slope of its worldline. The worldline of a free object is always straight. Its graph in a plot of its Minkowski coordinates is a straight line. R. H. Gowdy (VCU) General Relativity 03/06 4 / 26

19 The Equivalence Principal: Fictitious Forces Drop an object inside an accelerating rocket and it will appear to accelerate toward the oor as if acted upon by an invisible downward force. An outside observer will claim that this force is ctitious because it is really the oor of the laboratory that accelerates up toward the object. R. H. Gowdy (VCU) General Relativity 03/06 5 / 26

20 The Equivalence Principal: Gravity = Acceleration. An observer inside a (su ciently small) rocket cannot tell whether the rocket is accelerating through empty space or at rest in a gravitational eld as here. The inertial reference frames are freely falling. R. H. Gowdy (VCU) General Relativity 03/06 6 / 26

21 Curvature: How Straight Lines Can Be Curved These airplanes start out on parallel headings and each y straight according to local maps. Any map large enough to show their whole trips will have their paths curving toward each other. R. H. Gowdy (VCU) General Relativity 03/06 7 / 26

22 Curved Spacetime: Dust in a Falling Elevator A cloud of dust particles is initially at rest relative to a freely falling laboratory. Their world-lines start out parallel but curve toward (and also away from) each other, like the airplanes on the curved surface of the Earth. R. H. Gowdy (VCU) General Relativity 03/06 8 / 26

23 Curved Spacetime: A common misunderstanding The analogy between the airplane paths and curved spacetime has a problem. R. H. Gowdy (VCU) General Relativity 03/06 9 / 26

24 Curved Spacetime: A common misunderstanding The analogy between the airplane paths and curved spacetime has a problem. The airplane paths are drawn at an instant of time and re ect the space curvature of the Earth s surface. R. H. Gowdy (VCU) General Relativity 03/06 9 / 26

25 Curved Spacetime: A common misunderstanding The analogy between the airplane paths and curved spacetime has a problem. The airplane paths are drawn at an instant of time and re ect the space curvature of the Earth s surface. The dust particle worldlines, initially at rest, start out in the time direction. R. H. Gowdy (VCU) General Relativity 03/06 9 / 26

26 Curved Spacetime: A common misunderstanding The analogy between the airplane paths and curved spacetime has a problem. The airplane paths are drawn at an instant of time and re ect the space curvature of the Earth s surface. The dust particle worldlines, initially at rest, start out in the time direction. Pictures showing gravity as a result of curved space are misleading. R. H. Gowdy (VCU) General Relativity 03/06 9 / 26

27 Curved Spacetime: A common misunderstanding The analogy between the airplane paths and curved spacetime has a problem. The airplane paths are drawn at an instant of time and re ect the space curvature of the Earth s surface. The dust particle worldlines, initially at rest, start out in the time direction. Pictures showing gravity as a result of curved space are misleading. Space is indeed curved and that has e ects, but only for objects moving at high speed. R. H. Gowdy (VCU) General Relativity 03/06 9 / 26

28 Curved Spacetime: A common misunderstanding The analogy between the airplane paths and curved spacetime has a problem. The airplane paths are drawn at an instant of time and re ect the space curvature of the Earth s surface. The dust particle worldlines, initially at rest, start out in the time direction. Pictures showing gravity as a result of curved space are misleading. Space is indeed curved and that has e ects, but only for objects moving at high speed. The Newtonian gravity that we mostly experience at low speeds is due to curvature in the time direction. R. H. Gowdy (VCU) General Relativity 03/06 9 / 26

29 Einstein s Field Equations: What Newton s Theory Says The acceleration of a freely falling particle is given by the gradient of a potential. a i = φ x i R. H. Gowdy (VCU) General Relativity 03/06 10 / 26

30 Einstein s Field Equations: What Newton s Theory Says The acceleration of a freely falling particle is given by the gradient of a potential. a i = φ x i For the acceleration of a freely falling particle relative to a freely falling laboratory, expand the potential in a Taylor series around the center of mass of the lab and subtract out the gradient term. 1 a i = x 2 φ i r s 2 x r x s x r x s = j 2 φ x i x j x j ~a = 2 φ ~x R. H. Gowdy (VCU) General Relativity 03/06 10 / 26

31 Einstein s Field Equations: What Newton s Theory Says The acceleration of a freely falling particle is given by the gradient of a potential. a i = φ x i For the acceleration of a freely falling particle relative to a freely falling laboratory, expand the potential in a Taylor series around the center of mass of the lab and subtract out the gradient term. 1 a i = x 2 φ i r s 2 x r x s x r x s = j ~a = 2 φ ~x 2 φ x i x j x j In vacuum, Newton s inverse square law of gravity is equivalent to Tr 2 φ = r 2 φ = 0 or, the tidal force matrix, 2 φ is trace-free. R. H. Gowdy (VCU) General Relativity 03/06 10 / 26

32 Einstein s Field Equations: In vacuum Require that spacetime be curved in such a way that every small, freely falling laboratory will see a trace-free tidal force matrix. R. H. Gowdy (VCU) General Relativity 03/06 11 / 26

33 Einstein s Field Equations: In vacuum Require that spacetime be curved in such a way that every small, freely falling laboratory will see a trace-free tidal force matrix. In each laboratory, particles that move slowly in that lab obey Newtonian gravity. R. H. Gowdy (VCU) General Relativity 03/06 11 / 26

34 Einstein s Field Equations: In vacuum Require that spacetime be curved in such a way that every small, freely falling laboratory will see a trace-free tidal force matrix. In each laboratory, particles that move slowly in that lab obey Newtonian gravity. In each moving laboratory, the trace of the tidal force matrix is a component of the Ricci curvature tensor of spacetime. R. H. Gowdy (VCU) General Relativity 03/06 11 / 26

35 Einstein s Field Equations: In vacuum Require that spacetime be curved in such a way that every small, freely falling laboratory will see a trace-free tidal force matrix. In each laboratory, particles that move slowly in that lab obey Newtonian gravity. In each moving laboratory, the trace of the tidal force matrix is a component of the Ricci curvature tensor of spacetime. For all laboratories to see vanishing trace tidal forces, the spacetime must obey Ricci = 0 There are actually ten equations here. They make up Einstein s Vacuum Field equations and follow directly and simply from Newtonian Gravity and the Theory of Relativity. R. H. Gowdy (VCU) General Relativity 03/06 11 / 26

36 Einstein s Field Equations: When Matter is Present Where there is a mass-energy density ρ the Newtonian potential obeys r 2 φ = 4πG ρ R. H. Gowdy (VCU) General Relativity 03/06 12 / 26

37 Einstein s Field Equations: When Matter is Present Where there is a mass-energy density ρ the Newtonian potential obeys r 2 φ = 4πG ρ A rst guess at the equation that governs the curvature of spacetime would be Ricci = something involving mass-energy = 4πG (stress-energy tensor) R. H. Gowdy (VCU) General Relativity 03/06 12 / 26

38 Einstein s Field Equations: When Matter is Present Where there is a mass-energy density ρ the Newtonian potential obeys r 2 φ = 4πG ρ A rst guess at the equation that governs the curvature of spacetime would be Ricci = something involving mass-energy = 4πG (stress-energy tensor) Unfortunately the stress-energy tensor must obey an energy-momentum conservation law, while the Ricci tensor must obey the Bianchi Identity. R. H. Gowdy (VCU) General Relativity 03/06 12 / 26

39 Einstein s Field Equations: When Matter is Present Where there is a mass-energy density ρ the Newtonian potential obeys r 2 φ = 4πG ρ A rst guess at the equation that governs the curvature of spacetime would be Ricci = something involving mass-energy = 4πG (stress-energy tensor) Unfortunately the stress-energy tensor must obey an energy-momentum conservation law, while the Ricci tensor must obey the Bianchi Identity. The Bianchi Identity and the conservation law are not compatible, so the rst guess does not work. R. H. Gowdy (VCU) General Relativity 03/06 12 / 26

40 Einstein s Field Equations: When Matter is Present Where there is a mass-energy density ρ the Newtonian potential obeys r 2 φ = 4πG ρ A rst guess at the equation that governs the curvature of spacetime would be Ricci = something involving mass-energy = 4πG (stress-energy tensor) Unfortunately the stress-energy tensor must obey an energy-momentum conservation law, while the Ricci tensor must obey the Bianchi Identity. The Bianchi Identity and the conservation law are not compatible, so the rst guess does not work. The stress-energy needs a correction term that mixes mass-energy density and pressure. R. H. Gowdy (VCU) General Relativity 03/06 12 / 26

41 Einstein s Field Equations: When Matter is Present Where there is a mass-energy density ρ the Newtonian potential obeys r 2 φ = 4πG ρ A rst guess at the equation that governs the curvature of spacetime would be Ricci = something involving mass-energy = 4πG (stress-energy tensor) Unfortunately the stress-energy tensor must obey an energy-momentum conservation law, while the Ricci tensor must obey the Bianchi Identity. The Bianchi Identity and the conservation law are not compatible, so the rst guess does not work. The stress-energy needs a correction term that mixes mass-energy density and pressure. The local content of Newton s theory of gravity has to be modi ed. R. H. Gowdy (VCU) General Relativity 03/06 12 / 26

42 Einstein s Field Equations: Pressure attracts (but usually not very much) The simplest consistent coupling of matter to spacetime curvature is Ricci = 8πG (trace-reversed stress-energy tensor) The corresponding Newtonian eld equation is r 2 φ = 4πG ρ + 3 p c 2 R. H. Gowdy (VCU) General Relativity 03/06 13 / 26

43 Einstein s Field Equations: Pressure attracts (but usually not very much) The simplest consistent coupling of matter to spacetime curvature is Ricci = 8πG (trace-reversed stress-energy tensor) The corresponding Newtonian eld equation is r 2 φ = 4πG ρ + 3 p c 2 For ordinary pressures, the correction is very small because c 2 is very big. R. H. Gowdy (VCU) General Relativity 03/06 13 / 26

44 Einstein s Field Equations: Pressure attracts (but usually not very much) The simplest consistent coupling of matter to spacetime curvature is Ricci = 8πG (trace-reversed stress-energy tensor) The corresponding Newtonian eld equation is r 2 φ = 4πG ρ + 3 p c 2 For ordinary pressures, the correction is very small because c 2 is very big. However, if an object becomes massive enough for its central pressure to dominate its gravity, a paradox can result: R. H. Gowdy (VCU) General Relativity 03/06 13 / 26

45 Einstein s Field Equations: Pressure attracts (but usually not very much) The simplest consistent coupling of matter to spacetime curvature is Ricci = 8πG (trace-reversed stress-energy tensor) The corresponding Newtonian eld equation is r 2 φ = 4πG ρ + 3 p c 2 For ordinary pressures, the correction is very small because c 2 is very big. However, if an object becomes massive enough for its central pressure to dominate its gravity, a paradox can result: Instead of preventing the object from collapsing, increasing central pressure can accelerate the collapse. R. H. Gowdy (VCU) General Relativity 03/06 13 / 26

46 Einstein s Field Equations: Pressure attracts (but usually not very much) The simplest consistent coupling of matter to spacetime curvature is Ricci = 8πG (trace-reversed stress-energy tensor) The corresponding Newtonian eld equation is r 2 φ = 4πG ρ + 3 p c 2 For ordinary pressures, the correction is very small because c 2 is very big. However, if an object becomes massive enough for its central pressure to dominate its gravity, a paradox can result: Instead of preventing the object from collapsing, increasing central pressure can accelerate the collapse. That is one way to understand how a star can collapse to form a black hole. No physical "sti ening" process can stop the collapse because increasing the central pressure just makes things worse. R. H. Gowdy (VCU) General Relativity 03/06 13 / 26

47 Describing Spacetime: Coordinates Label events by arbitrary coordinates: R. H. Gowdy (VCU) General Relativity 03/06 14 / 26

48 Describing Spacetime: Coordinates Label events by arbitrary coordinates: An event P is labeled by the four numbers x 0 (P), x 1 (P), x 2 (P), x 3 (P) R. H. Gowdy (VCU) General Relativity 03/06 14 / 26

49 Describing Spacetime: Coordinates Label events by arbitrary coordinates: An event P is labeled by the four numbers x 0 (P), x 1 (P), x 2 (P), x 3 (P) These coordinates can cover an extensive region of spacetime, but may not be as regular as we would like. R. H. Gowdy (VCU) General Relativity 03/06 14 / 26

50 Describing Spacetime: Coordinates Label events by arbitrary coordinates: An event P is labeled by the four numbers x 0 (P), x 1 (P), x 2 (P), x 3 (P) These coordinates can cover an extensive region of spacetime, but may not be as regular as we would like. The worldlines of freely falling objects usually look curved in these coordinates. R. H. Gowdy (VCU) General Relativity 03/06 14 / 26

51 Describing Spacetime: Local Frames Use the procedures of Special Relativity to de ne regular Minkowski coordinates, t, x, y, z near each event. R. H. Gowdy (VCU) General Relativity 03/06 15 / 26

52 Describing Spacetime: Local Frames Use the procedures of Special Relativity to de ne regular Minkowski coordinates, t, x, y, z near each event. Near each event, we have a choice of Minkowski coordinates, which are regular there, but not elsewhere, and arbitrary coordinates, which label an extensive region but may not be regular anywhere. R. H. Gowdy (VCU) General Relativity 03/06 15 / 26

53 Describing Spacetime: Local Frames Use the procedures of Special Relativity to de ne regular Minkowski coordinates, t, x, y, z near each event. Near each event, we have a choice of Minkowski coordinates, which are regular there, but not elsewhere, and arbitrary coordinates, which label an extensive region but may not be regular anywhere. Freely falling objects have worldlines that look straight in these local Minkowski frames. R. H. Gowdy (VCU) General Relativity 03/06 15 / 26

54 Describing Spacetime: Frame- elds At each event, relate small changes in the Minkowski coordinates constructed from that event to small changes in the arbitrary coordinates. 0 t x y z 1 C A = 0 f (0) 0 f (0) 1 f (0) 2 f (0) 3 f (1) 0 f (1) 1 f (1) 2 f (1) 3 f (2) 0 f (2) 1 f (2) 2 f (2) 3 f (3) 0 f (3) 1 f (3) 2 f (3) C B x 0 x 1 x 3 x 4 1 C A R. H. Gowdy (VCU) General Relativity 03/06 16 / 26

55 Describing Spacetime: Frame- elds At each event, relate small changes in the Minkowski coordinates constructed from that event to small changes in the arbitrary coordinates. 0 t x y z 1 C A = 0 f (0) 0 f (0) 1 f (0) 2 f (0) 3 f (1) 0 f (1) 1 f (1) 2 f (1) 3 f (2) 0 f (2) 1 f (2) 2 f (2) 3 f (3) 0 f (3) 1 f (3) 2 f (3) C B x 0 x 1 x 3 x 4 At each event labeled by the arbitrary coordinates, there is a matrix of frame coe cients 0 f (0) 0 f (0) 1 f (0) 2 f (0) 1 3 [f ] = B f (1) 0 f (1) 1 f (1) 2 f (1) 3 f (2) 0 f (2) 1 f (2) 2 f (2) A 3 f (3) 0 f (3) 1 f (3) 2 f (3) 3 that o ers direct access to all of the local laws of physics as they are stated in a Minkowski reference frame. R. H. Gowdy (VCU) General Relativity 03/06 16 / 26 1 C A

56 Describing Spacetime: The Metric Tensor In Special relativity, the proper time interval between two events is related to the di erences in Minkowski coordinates by ( τ) 2 = ( t) 2 ( x) 2 ( y) 2 ( z) 2 R. H. Gowdy (VCU) General Relativity 03/06 17 / 26

57 Describing Spacetime: The Metric Tensor In Special relativity, the proper time interval between two events is related to the di erences in Minkowski coordinates by ( τ) 2 = ( t) 2 ( x) 2 ( y) 2 ( z) 2 Now we can express the proper time interval in terms of arbitrary coordinate di erences ( τ) 2 = = 3 3 α=0 β=0 3 3 α=0 β=0 f (0) αf (0) k x α x β f (0) αf (0) β i=1 α=0 β=0 3 f (i) αf (i) β i=1! f (i) αf (i) β x α x β x α x β R. H. Gowdy (VCU) General Relativity 03/06 17 / 26

58 Describing Spacetime: The Metric Tensor In Special relativity, the proper time interval between two events is related to the di erences in Minkowski coordinates by ( τ) 2 = ( t) 2 ( x) 2 ( y) 2 ( z) 2 Now we can express the proper time interval in terms of arbitrary coordinate di erences ( τ) 2 = = 3 3 α=0 β=0 3 3 α=0 β=0 f (0) αf (0) k x α x β f (0) αf (0) β i=1 α=0 β=0 3 f (i) αf (i) β i=1! f (i) αf (i) β x α x β x α x β The combinations of frame coe cients that appear here are the spacetime metric tensor components: g αβ = f (0) αf (0) β 3 f (i) αf (i) β i=1 R. H. Gowdy (VCU) General Relativity 03/06 17 / 26

59 Describing Spacetime: Spherical Coordinates Here is Minkowski spacetime with a frame- eld adapted to spherical coordinates. R. H. Gowdy (VCU) General Relativity 03/06 18 / 26

60 Describing Spacetime: Spherical Coordinates Here is Minkowski spacetime with a frame- eld adapted to spherical coordinates. Small changes in the frame coordinates, t, x, y, z, are related to changes in the general coordinates by dt = d t, dz = dr, dx = rdθ, dy = r sin θdφ R. H. Gowdy (VCU) General Relativity 03/06 18 / 26

61 Solving the Field Equations: Near a star (or a black hole) Einstein published the correct eld equations for spacetime in R. H. Gowdy (VCU) General Relativity 03/06 19 / 26

62 Solving the Field Equations: Near a star (or a black hole) Einstein published the correct eld equations for spacetime in They are a system of partial di erential equations for the metric tensor components (or equivalently the frame coe cients) as functions of arbitrary coordinates. R. H. Gowdy (VCU) General Relativity 03/06 19 / 26

63 Solving the Field Equations: Near a star (or a black hole) Einstein published the correct eld equations for spacetime in They are a system of partial di erential equations for the metric tensor components (or equivalently the frame coe cients) as functions of arbitrary coordinates. Karl Schwarzschild found the general spherically symmetric solution of the equations in Here it is in terms of the local Minkowski coordinates t, x, y, z. r 2Gm dt = 1 c 2 r d 1 t, dz = q dr 2Gm 1 c 2 r dx = rdθ, dy = r sin θdφ R. H. Gowdy (VCU) General Relativity 03/06 19 / 26

64 Solving the Field Equations: Near a star (or a black hole) Einstein published the correct eld equations for spacetime in They are a system of partial di erential equations for the metric tensor components (or equivalently the frame coe cients) as functions of arbitrary coordinates. Karl Schwarzschild found the general spherically symmetric solution of the equations in Here it is in terms of the local Minkowski coordinates t, x, y, z. r 2Gm dt = 1 c 2 r d 1 t, dz = q dr 2Gm 1 c 2 r dx = rdθ, dy = r sin θdφ Notice that something odd happens when r = 2Gm c 2. R. H. Gowdy (VCU) General Relativity 03/06 19 / 26

65 Solving the Field Equations: Near a star (or a black hole) Einstein published the correct eld equations for spacetime in They are a system of partial di erential equations for the metric tensor components (or equivalently the frame coe cients) as functions of arbitrary coordinates. Karl Schwarzschild found the general spherically symmetric solution of the equations in Here it is in terms of the local Minkowski coordinates t, x, y, z. r 2Gm dt = 1 c 2 r d 1 t, dz = q dr 2Gm 1 c 2 r dx = rdθ, dy = r sin θdφ Notice that something odd happens when r = 2Gm. c 2 That value is called the Schwarzschild Radius. If m is the mass of the Earth, it is about a centimeter. R. H. Gowdy (VCU) General Relativity 03/06 19 / 26

66 Solving the Field Equations: Near a star (or a black hole) Einstein published the correct eld equations for spacetime in They are a system of partial di erential equations for the metric tensor components (or equivalently the frame coe cients) as functions of arbitrary coordinates. Karl Schwarzschild found the general spherically symmetric solution of the equations in Here it is in terms of the local Minkowski coordinates t, x, y, z. r 2Gm dt = 1 c 2 r d 1 t, dz = q dr 2Gm 1 c 2 r dx = rdθ, dy = r sin θdφ Notice that something odd happens when r = 2Gm. c 2 That value is called the Schwarzschild Radius. If m is the mass of the Earth, it is about a centimeter. It took almost 50 years for its signi cance to become clear. R. H. Gowdy (VCU) General Relativity 03/06 19 / 26

67 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

68 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is The same everywhere and in all directions. R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

69 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is The same everywhere and in all directions. Has constant-time surfaces that are Euclidean. R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

70 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is The same everywhere and in all directions. Has constant-time surfaces that are Euclidean. This type of universe can be described by a global time coordinate t and global Cartesian coordinates x, ȳ, z. R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

71 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is The same everywhere and in all directions. Has constant-time surfaces that are Euclidean. This type of universe can be described by a global time coordinate t and global Cartesian coordinates x, ȳ, z. A local Minkowski coordinate system t, x, y, z in such a universe could be connected to the global coordinates by the relations dt = d t, dx = a ( t) d x, dy = a ( t) dȳ, dz = a ( t) d z R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

72 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is The same everywhere and in all directions. Has constant-time surfaces that are Euclidean. This type of universe can be described by a global time coordinate t and global Cartesian coordinates x, ȳ, z. A local Minkowski coordinate system t, x, y, z in such a universe could be connected to the global coordinates by the relations dt = d t, dx = a ( t) d x, dy = a ( t) dȳ, dz = a ( t) d z Plug these frame coe cients into Einstein s eld equations along with the equations that govern the density and pressure of the matter content of the universe. R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

73 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is The same everywhere and in all directions. Has constant-time surfaces that are Euclidean. This type of universe can be described by a global time coordinate t and global Cartesian coordinates x, ȳ, z. A local Minkowski coordinate system t, x, y, z in such a universe could be connected to the global coordinates by the relations dt = d t, dx = a ( t) d x, dy = a ( t) dȳ, dz = a ( t) d z Plug these frame coe cients into Einstein s eld equations along with the equations that govern the density and pressure of the matter content of the universe. Get an ordinary di erential equation for the function a ( t) that indicates how the universe is expanding. R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

74 Solving the Field Equations: For the universe The simplest model of the universe assumes that, on the largest scales, spacetime is The same everywhere and in all directions. Has constant-time surfaces that are Euclidean. This type of universe can be described by a global time coordinate t and global Cartesian coordinates x, ȳ, z. A local Minkowski coordinate system t, x, y, z in such a universe could be connected to the global coordinates by the relations dt = d t, dx = a ( t) d x, dy = a ( t) dȳ, dz = a ( t) d z Plug these frame coe cients into Einstein s eld equations along with the equations that govern the density and pressure of the matter content of the universe. Get an ordinary di erential equation for the function a ( t) that indicates how the universe is expanding. There are more complicated models with non-euclidean constant-time surfaces, but it is this simple Euclidean one that best ts the data. R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

75 Solving the Field Equations: For gravitational waves To represent small ripples in spacetime, use a matrix of frame coe cients [f ] = [1] + ε [h] where ε is a parameter that ranges from 0 to 1. R. H. Gowdy (VCU) General Relativity 03/06 21 / 26

76 Solving the Field Equations: For gravitational waves To represent small ripples in spacetime, use a matrix of frame coe cients [f ] = [1] + ε [h] where ε is a parameter that ranges from 0 to 1. Expand Einstein s Field Equations in powers of ε and just keep the rst order terms. The result is called the linearized theory. R. H. Gowdy (VCU) General Relativity 03/06 21 / 26

77 Solving the Field Equations: For gravitational waves To represent small ripples in spacetime, use a matrix of frame coe cients [f ] = [1] + ε [h] where ε is a parameter that ranges from 0 to 1. Expand Einstein s Field Equations in powers of ε and just keep the rst order terms. The result is called the linearized theory. Impose the coordinate condition that each extended coordinate function should obey the wave equation in the curved spacetime: x = ȳ = z = t and nd that the frame coe cients then obey a wave equation as well as some constraints. [h] = 0 R. H. Gowdy (VCU) General Relativity 03/06 21 / 26

78 Solving the Field Equations: For gravitational waves Solve the constraints and the wave equation the way we always do by substituting [h] = Re [a] e i(~ k~r ωt) R. H. Gowdy (VCU) General Relativity 03/06 22 / 26

79 Solving the Field Equations: For gravitational waves Solve the constraints and the wave equation the way we always do by substituting [h] = Re [a] e i(~ k~r ωt) For a wave propagating in the z direction, there are two independent solutions: [h] + = Re [a + ] e ik( z ct) ik( z, [h] = Re [a ] e ct) where [a + ] = a a C A, [a ] = a 0 0 a C A R. H. Gowdy (VCU) General Relativity 03/06 22 / 26

80 Solving the Field Equations: For gravitational waves Here are the changes in coordinates. R. H. Gowdy (VCU) General Relativity 03/06 23 / 26

81 Solving the Field Equations: For gravitational waves Here are the changes in coordinates. For both polarizations, nothing changes in the direction of propagation: dt = d t, dz = d z For the + polarization: For the polarization: dx = (1 + εa) d x, dy = (1 εa) dȳ dx = d x + εadȳ, dy = dȳ + εad x R. H. Gowdy (VCU) General Relativity 03/06 23 / 26

82 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

83 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

84 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

85 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

86 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

87 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

88 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

89 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object Tests the Schwarzschild metric in the nonlinear regime R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

90 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object Tests the Schwarzschild metric in the nonlinear regime Provided rst observational evidence of gravitational waves R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

91 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object Tests the Schwarzschild metric in the nonlinear regime Provided rst observational evidence of gravitational waves The Microwave Background Radiation R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

92 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object Tests the Schwarzschild metric in the nonlinear regime Provided rst observational evidence of gravitational waves The Microwave Background Radiation Direct observational evidence of Big Bang Cosmology R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

93 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object Tests the Schwarzschild metric in the nonlinear regime Provided rst observational evidence of gravitational waves The Microwave Background Radiation Direct observational evidence of Big Bang Cosmology High Precision measurements of the average matter content of the universe R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

94 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object Tests the Schwarzschild metric in the nonlinear regime Provided rst observational evidence of gravitational waves The Microwave Background Radiation Direct observational evidence of Big Bang Cosmology High Precision measurements of the average matter content of the universe Supports the In ation model of the very early universe (another story) R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

95 Testing the Predictions of General Relativity Solar System Tests (Testing the Schwarzschild metric) Include light bending near the Sun, the Perihelion Precession of Mercury Phase shifting of quasar radio signals past the Sun Laser ranging to retrore ectors on the Moon Radar ranging experiments and space probes The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize) An accurate clock (pulsar) in close orbit around a collapsed object Tests the Schwarzschild metric in the nonlinear regime Provided rst observational evidence of gravitational waves The Microwave Background Radiation Direct observational evidence of Big Bang Cosmology High Precision measurements of the average matter content of the universe Supports the In ation model of the very early universe (another story) Indicates the universe is mostly "dark energy," which is matter with negative pressure. R. H. Gowdy (VCU) General Relativity 03/06 24 / 26

96 Testing the Predictions of General Relativity Astronomical Phenomena R. H. Gowdy (VCU) General Relativity 03/06 25 / 26

97 Testing the Predictions of General Relativity Astronomical Phenomena Hubble Expansion of the Universe R. H. Gowdy (VCU) General Relativity 03/06 25 / 26

98 Testing the Predictions of General Relativity Astronomical Phenomena Hubble Expansion of the Universe Phenomena only explainable as black holes R. H. Gowdy (VCU) General Relativity 03/06 25 / 26

99 Testing the Predictions of General Relativity Astronomical Phenomena Hubble Expansion of the Universe Phenomena only explainable as black holes The relative abundances of elements R. H. Gowdy (VCU) General Relativity 03/06 25 / 26

100 Testing the Predictions of General Relativity Astronomical Phenomena Hubble Expansion of the Universe Phenomena only explainable as black holes The relative abundances of elements Models of Neutron Stars and Supernovas R. H. Gowdy (VCU) General Relativity 03/06 25 / 26

101 The Search for Gravitational Radiation Hanford, WA Livingston, LA GEO600, Hannover VIRGO on the Arno AIGO in West Australia TAMA, Tokyo R. H. Gowdy (VCU) General Relativity 03/06 26 / 26