Einstein for Everyone Lecture 6: Introduction to General Relativity

Transcription

1 Einstein for Everyone Lecture 6: Introduction to General Relativity Dr. Erik Curiel Munich Center For Mathematical Philosophy Ludwig-Maximilians-Universität

2 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

3 Why a New Theory of Gravity? Einstein s Special Relativity Newtonian Minkowski spacetime Space and time observer dependent, replaced by invariant space-time interval Newtonian Gravity Incredibly empirically successful Force of gravity: - depends on spatial distance at a single instant of time - instantaneous interaction absolute simultaneity Challenge New theory of gravity compatible with special relativity?

4 Responses to the Challenge Einstein s Contemporaries (Poincaré, Minkowski, Max Abraham, Gustav Mie... ) - Reformulate gravity in Minkowski spacetime - Preserve special relativity, change theory of gravity Einstein s Approach - Relativity as an incomplete revolution - Change both special relativity and theory of gravity - Conceptual problems within Newtonian gravity - reformulate notion of relativistic spacetime - need to generalize notion of geometry

5 Responses to the Challenge Einstein s Contemporaries (Poincaré, Minkowski, Max Abraham, Gustav Mie... ) - Reformulate gravity in Minkowski spacetime - Preserve special relativity, change theory of gravity Einstein s Approach - Relativity as an incomplete revolution - Change both special relativity and theory of gravity - Conceptual problems within Newtonian gravity - reformulate notion of relativistic spacetime - need to generalize notion of geometry

6 Why Geometry? Einstein s rough and winding road ( ) 1905 Special relativity 1907 Happiest thought of my life (principle of equivalence) Equivalence between gravity and acceleration Need to extend relativity to accelerated frames 1909 Ehrenfest s Rotating Disk Acceleration Non-Euclidean Geometry Hole Argument 1915 General theory of relativity

7 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

8 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Abs Kepler s Laws Copernican Revolution Ptolemaic Hypothesis Copernican Hypothesis Images from Hevelius, Selenographica (1647) (courtesy of Trinity College, Cambridge)

9 Kepler s Laws Johannes Kepler ( ) Kepler s Innovations Orbit of Mars: Ellipse Motion of planets caused by sun, analogy with magnetism Kepler s New Astronomy (1609) Kepler s Laws 1 Planets move along an ellipse with the sun at one focus. 2 They sweep out equal areas in equal times. 3 The radius of the orbit a is related to the period P as P 2 a 3

10 Inertia and Acceleration Types of Motion Inertial Motion Motion in a straight line with uniform velocity (that is, covering equal distances in equal times). Accelerated Motion Change in velocity (speed up or slow down) or direction (e.g., rotation) Based on Newtonian space and time: Spatial Geometry: straight line; distances measured by measuring rods Time: time elapsed, measured by a clock Location over time: distance traveled over time

11 Inertia and Acceleration Newton s First Law Law I Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed upon it.

12 Inertia and Acceleration Newton s Second Law Law II A change in motion is proportional to the motive force impressed and takes place in the direction of the straight line along which that force is impressed.

13 Inertia and Acceleration Force and Inertial Mass Modern formulation of second law: F = m i a Force F Measured by departure from inertial motion Treated abstractly, quantitatively Examples: impact, attraction (magnetism, gravitation), dissipation (friction), tension (oscillating string),... Inertial Mass m i Intrinsic property of a body Measures how much force is required to accelerate a body Not equivalent to weight

14 Gravity Newtonian Gravity Attractive force between interacting bodies M, m: F = G Mgmg r 2 Dependence on Distance - F 1 r 2 - Move bodies twice as far apart, force decreases by 1 4 Dependence on Masses - Force depends on gravitational masses of both interacting bodies

15 Gravity Newton s Argument for Universal Gravitation 1 Kepler s Laws Force F 1 r 2 - Kepler s laws hold for planets and satellites 2 Moon Test : this force is gravity! - Compare force on moon to force on falling bodies near Earth s surface 3 Dependence on Mass - Pendulum Experiments 4 Conclusion: universal, mutual attractive force

16 Gravity Galileo on Freely Falling Bodies Bodies fall in the same way regardless of composition Two separate concepts of mass 1 Inertial mass: F = m i a 2 Gravitational mass: F = G Mgmg r 2 If m i = m g, then Galileo s result is true!

17 Gravity Objections to Newton s Theory (ca. 1907) 1 Problems due to Special Relativity - Space and time no longer invariant - Instantaneous interaction 2 Epistemological Defect in Newton s theory - Why are inertial and gravitational mass equal? - Absolute space 3 Empirical Problems - Motion of Mercury - (Lunar motion, motion of Venus)

18 Conservatives vs. Einstein Relativity Meets Gravity Problems due to Special Relativity Spatial distance between two bodies observer-dependent Time at which force acts observer-dependent Conservative Response Reformulate gravity in terms of space-time distance Minkowski, Poincaré, Abraham, Mie, Nördström: several possibilities, fairly straightforward modification

19 Conservatives vs. Einstein Galileo s Treatment of Free Fall Bodies fall in the same way regardless of composition (or amount of energy) Consequence of m i = m g Implies that time of fall is the same regardless of horizontal velocity

20 Conservatives vs. Einstein... Conflicts with Special Relativity! - Observer A: bodies all land simultaneously - Observer B: bodies cannot all land simultaneously Conclusion: Galileo was wrong!?

21 Conservatives vs. Einstein Einstein vs. the Conservative Approach Conservative Response Galileo s idea was wrong, special relativity is correct! (Compatible with empirical evidence as long as Galileo s claim holds approximately) Einstein s Response Galileo s idea was correct, special relativity is wrong! Galileo s idea: crucial insight that should be preserved Need to extend relativity theory, develop a new theory

22 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

23 Extending Relativity Principle of Relativity Redux Principle of Relativity All observers in inertial motion (inertial observers) see the same laws of physics. Einstein s Questions (1907): Does an extended version of this principle hold for accelerated observers? How does extending the principle help us to understand gravity?

24 Extending Relativity Newton s Hint Relativity for Accelerated Frames? If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces. (Corollary 6 to Laws of Motion)

25 Extending Relativity Newton s Hint Locally freely falling frame (uniform acceleration) equivalent to inertial frame! - Qualification: Acceleration directed along parallel lines. Usually this will be true only locally as an approximation. - Status of the distinction between gravity and inertia? - Another theoretical asymmetry which does not appear to be inherent in the phenomena?

26 Extending Relativity Relativity of Gravity and Acceleration From Janssen, No Success like Failure...

27 Extending Relativity Relativity of Gravity and Acceleration

28 Extending Relativity Relativity of Gravity and Acceleration Relativity Extended to Acceleration Either observer can claim to be at rest, disagree about whether there is gravity (I) and (II) can be accounted for with gravity or with acceleration

29 Extending Relativity Einstein s Equivalence Principle 1907 Gravity and acceleration are physically indistinguishable - But this holds only locally - Not all cases of acceleration can be replaced by gravitational field 1910s Various different formulations of the idea 1915 Relativity of gravity - Inertia and gravity are aspects of the same underlying thing; breaks down into components relative to observer - Need generalized geometry to describe new notion of straight line

30 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

31 Gravitational Time Dilation Einstein (1907) Strategy - Consider accelerated observers in special relativity, use reasoning regarding relativity of simultaneity - invoke Principle of Equivalence for connection with gravity

32 Gravitational Time Dilation Uniform Acceleration

33 Gravitational Time Dilation Time Dilation

34 Light Bending: Trajectory and Speed of Light Trajectory of Light

35 Light Bending: Trajectory and Speed of Light Speed of Light

36 Light Bending: Trajectory and Speed of Light Speed of Light

37 Light Bending: Trajectory and Speed of Light Summary: Using the Equivalence Principle Einstein (1912): results for static gravity 1 Light bends in a gravitational field 2 Clocks run slow in a gravitational field (static means not changing with time)

38 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

39 Guiding Principles Equivalence Principle - Freely falling frame (gravity + inertia) equivalent to inertial frame - Qualification: true only locally, does not apply to all cases - Einstein s insight: theory should treat inertia and gravity as aspects of the same thing, unity of essence Mach s Principle - Criticize Newtonian absolute space as basis for defining inertial motions - Inertia due to interaction with other bodies

40 Epistemological Defect in Newton s Theory What is the distinction between inertial and non-inertial motion? (Why are some states of motion singled out as inertial?) Newton s Answer: motion defined with respect to space itself Mach and Einstein: motion defined with respect to other bodies

41 Dialogue from Einstein (1914) Two masses, close enough so that they interact. Consider looking along the line between them towards the starry night sky. Mach My masses carry out a motion, which is at least in part causally determined by the fixed stars. The law by which masses in my surroundings move is co-determined by the fixed stars. Newton The motion of your masses has nothing to do with the heaven of fixed stars; it is rather fully determined by the laws of mechanics entirely independently of the remaining masses. There is a space S in which these laws hold.

42 Mach But just as I could never be brought to believe in ghosts, so I cannot believe in this gigantic thing that you speak of and call space. I can neither see something like that nor conceive of it. Or should I think of your space S as a subtle net of bodies that the remaining things are all referred to? Then I can imagine a second such net S in addition to S, that is moving in an arbitrary manner relative to S (for example, rotating). Do your equations also hold at the same time with respect to S? Newton No Mach But how do the masses know which space S, S, etc., with respect to which they should move according to your equations, how do they recognize the space or spaces they orient themselves with respect to?... I will take, for the time being, your privileged spaces as an idle fabrication, and stay with my conception, that the sphere of fixed stars co-determines the mechanical behavior of my test masses.

43 Epistemological Defect What causes the objects to move as they do? - Newtonian : in space S the laws of physics hold. Apply the laws predict motion of the system. Machian criticisms - What justifies the choice of S, rather than S? - This space is unobservable! (Inappropriate to invoke invisible causes )

44 Mach s Principle Alternative to Newton s appeal to absolute space - Define inertia with respect to distant stars :... the sphere of fixed stars co-determines the mechanical behavior of my test masses Connection with Equivalence Principle - Equivalence principle breaks down distinction between inertial and accelerated motion - Inertia and gravity linked

45 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

46 Euclid s Elements Geometry pre-euclid - Assortment of accepted results, e.g. Pythagoras s theorem - How do these results relate to each other? How does one give a convincing argument in favor of such results? What would make a good proof? Euclid s Elements - Deductive structure - Starting points: definitions, axioms, postulates - Proof: show that other claims follow from definitions - Build up to more complicated proofs step-by-step

47 Deductive Structure Deductive Structure of Geometry Definitions 23 geometrical terms D 1 A point is that which has no part. D 2 A line is breadthless length.... D 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Axioms General principles of reasoning, also called common notions A 1 Things which equal the same thing also equal one another.... Postulates Regarding possible geometrical constructions

48 Deductive Structure Euclid s Five Postulates 1. To draw a straight line from any point to any point. 2. To produce a limited straight line in a straight line. 3. To describe a circle with any center and distance. 4. All right angles are equal to one another. 5. If one straight line falling on two straight lines makes the interior angles in the same direction less than two right angles, then the two straight lines, if produced in infinitum, meet one another in that direction in which the angles less than two right angles are.

49 Deductive Structure Status of Geometry Exemplary case of demonstrative knowledge - Theorems based on clear, undisputed definitions and postulates - Clear deductive structure showing how theorems follow Philosophical questions - How is knowledge of this kind (synthetic rather than merely analytic) possible? - What is the subject matter of geometry? Why is geometry applicable to the real world?

50 Fifth Postulate Euclid s Fifth Postulate 5. If one straight line falling on two straight lines makes the interior angles in the same direction less than two right angles, then the two straight lines, if produced in infinitum, meet one another in that direction in which the angles less than two right angles are. 5-ONE Simpler, equivalent formulation: Given a line and a point not on the line, there is one line passing through the point parallel to the given line.

51 Fifth Postulate Significance of Postulate 5 Contrast with Postulates More complex, less obvious statement - Used to introduce parallel lines, extendability of constructions - Only axiom to refer to, rely on possibly infinite magnitudes Prove or dispense with Postulate 5? - Long history of attempts to prove Postulate 5 from other postulates, leads to independence proofs - Isolate the consequences of Postulate 5 - Saccheri (1733), Euclid Freed from Every Flaw: attempts to derive absurd consequences from denial of 5-ONE

52 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

53 Introduction Alternatives for Postulate 5 5-ONE Given a line and a point not on the line, there is one line passing through the point parallel to the given line. 5-NONE Given a line and a point not on the line, there are no lines passing through the point parallel to the given line. -MANY Given a line and a point not on the line, there are many lines passing through the point parallel to the given line.

54 Introduction Geometrical Construction for 5-NONE Reductio ad absurdum? Saccheri s approach: assuming 5-NONE or 5-MANY (and other postulates) leads to contradictions, so 5-ONE must be correct. Construction: assuming 5-NONE, construct triangles with a common line as base Results: sum of angles of a triangle > 180 ; circumference 2πR

55 Introduction Non-Euclidean Geometries Pre-1830 (Saccheri et al.) Study alternatives to find contradiction Prove a number of results for absurd geometries with 5-NONE, 5-MANY Nineteenth Century These are fully consistent alternatives to Euclid 5-NONE: spherical geometry 5-MANY: hyperbolic geometry

56 Introduction Hyperbolic Geometry

57 Introduction Consequences 5-??? What depends on choice of a version of postulate 5? - Procedure: Go back through Elements, trace dependence on 5-ONE Replace with 5-NONE or 5 -MANY and derive new results - Results: sum of angles of triangle 180, C 2πr,...

58 Spherical Geometry Geometry of 5-NONE What surface has the following properties? Pick an arbitrary point. Circles: - Nearby have C 2πR - As R increases, C < 2πR Angles sum to more than Euclidean case (for triangles, quadrilaterals, etc.) True for every point sphere

59 Hyperbolic Geometry Geometry of 5-MANY Properties of hyperboloid surface: Extra space Circumference > 2πR Angles sum to less than Euclidean case (for triangles, quadrilaterals, etc.)

60 Summary Status of these Geometries? How to respond to Saccheri et al., who thought a contradiction follows from 5-NONE or 5-MANY? Relative Consistency Proof If Euclidean geometry is consistent, then hyperbolic / spherical geometry is also consistent. Proof based on translation Euclidean non-euclidean

61 Summary Summary: Three Non-Euclidean Geometries Geometry Parallels Straight Lines Triangles Circles Euclidean 5-ONE C = 2πR Spherical 5-NONE finite > 180 C < 2πR Hyperbolic 5-MANY < 180 C > 2πR Common Assumptions Intrinsic geometry for surfaces of constant curvature. Further generalization (Riemann): drop this assumption!

62 1 Introduction to General Relativity 2 Newtonian Gravity Kepler s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein 3 Equivalence Principle Extending Relativity 4 Using the Equivalence Principle Gravitational Time Dilation Light Bending: Trajectory and Speed of

63 Intrinsic vs. Extrinsic Geometry on a Surface What does geometry of figures drawn on surface of a sphere mean? Intrinsic geometry - Geometry on the surface; measurements confined to the 2-dimensional surface Extrinsic geometry - Geometry of the surface as embedded in another space - 2-dimensional spherical surface in 3-dimensional space

64 Intrinsic vs. Extrinsic Geometry on a Surface What does geometry of figures drawn on surface of a sphere mean? Intrinsic geometry - Geometry on the surface; measurements confined to the 2-dimensional surface Extrinsic geometry - Geometry of the surface as embedded in another space - 2-dimensional spherical surface in 3-dimensional space

65 Intrinsic vs. Extrinsic Geometry on a Surface What does geometry of figures drawn on surface of a sphere mean? Intrinsic geometry - Geometry on the surface; measurements confined to the 2-dimensional surface Extrinsic geometry - Geometry of the surface as embedded in another space - 2-dimensional spherical surface in 3-dimensional space

66 Intrinsic vs. Extrinsic Importance of Being Intrinsic Extrinsic geometry useful... but limited in several ways: - Not all surfaces can be fully embedded in higher-dimensional space - Limits of visualization: 3-dimensional surface embedded in 4-dimensional space? So focus on intrinsic geometry instead

67 Intrinsic vs. Extrinsic Importance of Being Intrinsic Extrinsic geometry useful... but limited in several ways: - Not all surfaces can be fully embedded in higher-dimensional space - Limits of visualization: 3-dimensional surface embedded in 4-dimensional space? So focus on intrinsic geometry instead

68 Intrinsic vs. Extrinsic Importance of Being Intrinsic Extrinsic geometry useful... but limited in several ways: - Not all surfaces can be fully embedded in higher-dimensional space - Limits of visualization: 3-dimensional surface embedded in 4-dimensional space? So focus on intrinsic geometry instead

69 Curvature Curvature of a Line

70 Curvature Curvature of a Surface

71 Geodesic Deviation Intrinsic Characterization of Curvature Behavior of nearby initially parallel lines, reflects curvature

72 Geodesic Deviation Non-Euclidean Geometries Revisited Geometry Parallels Curvature Geodesic Deviation Euclidean 5-ONE zero constant Spherical 5-NONE positive converge Hyperbolic 5-MANY negative diverge Riemannian Geometry Curvature allowed to vary from point to point; link with geodesic deviation still holds.

73 Geodesic Deviation Non-Euclidean Geometries Revisited Geometry Parallels Curvature Geodesic Deviation Euclidean 5-ONE zero constant Spherical 5-NONE positive converge Hyperbolic 5-MANY negative diverge Riemannian Geometry Curvature allowed to vary from point to point; link with geodesic deviation still holds.