Einstein for Everyone Lecture 6: Introduction to General Relativity

Dr. Erik Curiel

Munich Center For Mathematical Philosophy Ludwig-Maximilians-Universit¨at Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Light

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

8 Riemannian Geometry Intrinsic vs. Extrinsic Curvature Geodesic Deviation Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Why a New Theory of Gravity?

Newtonian Gravity Einstein’s Special Relativity Incredibly empirically Newtonian ⇒ Minkowski successful spacetime Force of gravity: Space and time observer - depends on spatial dependent, replaced by distance at a single invariant space-time instant of time interval - instantaneous interaction ⇒ absolute simultaneity Challenge New theory of gravity compatible with special relativity? Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Responses to the Challenge

Einstein’s Contemporaries (Poincar´e,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” Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Responses to the Challenge

Einstein’s Contemporaries (Poincar´e,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” Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Why Geometry?

Einstein’s “rough and winding road” (1907-1915) 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 1912-13 Hole Argument 1915 General theory of relativity Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Light

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

8 Riemannian Geometry Intrinsic vs. Extrinsic Curvature Geodesic Deviation Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Kepler’s Laws Copernican Revolution

Ptolemaic Hypothesis Copernican Hypothesis

Images from Hevelius, Selenographica (1647) (courtesy of Trinity College, Cambridge) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Kepler’s Laws Johannes Kepler (1571-1630)

Kepler’s Innovations Orbit of Mars: Ellipse Motion of planets caused by sun, analogy with magnetism

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 Kepler’s New Astronomy to the period P as P 2 ∝ a3 (1609) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Inertia and Acceleration Force and Inertial Mass

Modern formulation of second law: F = mia

Force F Measured by departure Inertial Mass mi from inertial motion Intrinsic property of a Treated abstractly, body quantitatively Measures how much force Examples: impact, is required to accelerate a attraction (magnetism, body gravitation), dissipation Not equivalent to weight (friction), tension (oscillating string), . . . Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Gravity Newtonian Gravity

Attractive force between interacting bodies M, m: Mgmg F = G r2

Dependence on Distance 1 - F ∝ r2 1 - Move bodies twice as far apart, force decreases by 4 Dependence on Masses - Force depends on gravitational masses of both interacting bodies Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Gravity Newton’s Argument for Universal Gravitation

1 1 Kepler’s Laws → Force F ∝ r2 - 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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Gravity Galileo on Freely Falling Bodies

Bodies fall in the same way regardless of composition Two separate concepts of mass

1 Inertial mass: F = mia 2 Gravitational mass: Mg mg F = G r2 If mi = mg, then Galileo’s result is true! Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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, Poincar´e,Abraham, Mie, N¨ordstr¨om:several possibilities, fairly straightforward modiﬁcation Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Conservatives vs. Einstein Galileo’s Treatment of Free Fall

Bodies fall in the same way regardless of composition (or amount of energy) Consequence of mi = mg Implies that time of fall is the same regardless of horizontal velocity Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Conservatives vs. Einstein ... Conﬂicts with Special Relativity!

- Observer A: bodies all land simultaneously - Observer B: bodies cannot all land simultaneously Conclusion: Galileo was wrong!? Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Light

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

8 Riemannian Geometry Intrinsic vs. Extrinsic Curvature Geodesic Deviation Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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? Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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”? Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Extending Relativity Relativity of Gravity and Acceleration

From Janssen, “No Success like Failure...” Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Extending Relativity Relativity of Gravity and Acceleration Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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” Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

1 Introduction to General Relativity

2 Newtonian Gravity Kepler’s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein

3 Equivalence Principle Extending Relativity

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Gravitational Time Dilation Uniform Acceleration Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Gravitational Time Dilation Time Dilation Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Light Bending: Trajectory and Speed of Light Trajectory of Light Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Light Bending: Trajectory and Speed of Light Speed of Light Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Light Bending: Trajectory and Speed of Light Speed of Light Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

1 Introduction to General Relativity

2 Newtonian Gravity Kepler’s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein

3 Equivalence Principle Extending Relativity

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 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. 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 S0 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 S0? Newton No Mach But how do the masses know which “space” S, S0, 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. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 S0? - This “space” is unobservable! (Inappropriate to invoke “invisible causes”) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

1 Introduction to General Relativity

2 Newtonian Gravity Kepler’s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein

3 Equivalence Principle Extending Relativity

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

Euclid’s Elements

Geometry pre-Euclid Euclid’s Elements - Assortment of accepted - Deductive structure results, e.g. Pythagoras’s - Starting points: deﬁnitions, theorem axioms, postulates - How do these results relate to - Proof: show that other claims each other? How does one follow from deﬁnitions give a convincing argument in - Build up to more complicated favor of such results? What proofs step-by-step would make a good “proof”? Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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? Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Fifth Postulate Signiﬁcance of Postulate 5

Contrast with Postulates 1-4 - 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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

1 Introduction to General Relativity

2 Newtonian Gravity Kepler’s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein

3 Equivalence Principle Extending Relativity

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

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. 5-MANY Given a line and a point not on the line, there are many lines passing through the point parallel to the given line. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 6= 2πR Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Introduction Non-Euclidean Geometries

Nineteenth Century Pre-1830 (Saccheri et al.) These are fully consistent Study alternatives to ﬁnd alternatives to Euclid contradiction 5-NONE: spherical Prove a number of results geometry for “absurd” geometries 5-MANY: hyperbolic with 5-NONE, 5-MANY geometry Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Introduction Hyperbolic Geometry Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 6= 180◦, C 6= 2πr,... Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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.) Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Summary Summary: Three Non-Euclidean Geometries

Geometry Parallels Straight Lines Triangles Circles Euclidean 5-ONE ... 180◦ 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! Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

1 Introduction to General Relativity

2 Newtonian Gravity Kepler’s Laws Inertia and Acceleration Gravity Conservatives vs. Einstein

3 Equivalence Principle Extending Relativity

5 Rejection of Absolute Space

6 Euclidean Geometry Deductive Structure Fifth Postulate

7 non-Euclidean Geometry Introduction Spherical Geometry Hyperbolic Geometry Summary

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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 Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Curvature Curvature of a Line Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Curvature Curvature of a Surface Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

Geodesic Deviation Intrinsic Characterization of Curvature

Behavior of nearby initially parallel lines, reﬂects curvature Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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. Introduction to General Relativity Newtonian Gravity Equivalence Principle Using the Equivalence Principle Rejection of Absolute Space Euclidean Geometry non-Euclidean Geometry Riemannian Geometry

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.