Einstein for Everyone Lecture 6:Introduction to General Relativity

Dr. Erik Curiel

Munich Center For Mathematical PhilosophyLudwig-Maximilians-Universitat

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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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?

Einstein’s Special Relativity

Newtonian ⇒ Minkowskispacetime

Space and time observerdependent, replaced byinvariant space-timeinterval

Newtonian Gravity

Incredibly empiricallysuccessful

Force of gravity:

- depends on spatialdistance at a singleinstant of time

- 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 (Poincare, 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 (Poincare, 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 accelerationNeed 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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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 New Astronomy(1609)

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 withthe sun at one focus.

2 They sweep out equal areas inequal times.

3 The radius of the orbit a is relatedto the period P as P 2 ∝ a3

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, coveringequal 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 bymeasuring rodsTime: time elapsed, measured by a clockLocation 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 movinguniformly straight forward, except insofar as it is compelled tochange 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 impressedand takes place in the direction of the straight line along whichthat 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 departurefrom inertial motion

Treated abstractly,quantitatively

Examples: impact,attraction (magnetism,gravitation), dissipation(friction), tension(oscillating string), . . .

Inertial Mass mi

Intrinsic property of abody

Measures how much forceis required to accelerate abody

Not equivalent to weight

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:F = G

Mgmg

r2

Dependence on Distance

- F ∝ 1

r2

- Move bodies twice as far apart, force decreases by1

4Dependence on Masses

- Force depends on gravitational masses of both interactingbodies

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 Kepler’s Laws → Force F ∝ 1r2

- 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’ssurface

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 wayregardless of composition

Two separate concepts of mass

1 Inertial mass: F = mia2 Gravitational mass:

F = GMgmg

r2

If mi = mg, then Galileo’s resultis 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-dependentTime at which force acts observer-dependent

Conservative Response

Reformulate gravity in terms of space-time distanceMinkowski, Poincare, Abraham, Mie, Nordstrom: severalpossibilities, 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 sameway regardless ofcomposition (or amountof energy)

Consequence ofmi = mg

Implies that time offall is the sameregardless ofhorizontal 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: bodiesall landsimultaneously

- Observer B: bodiescannot all landsimultaneously

Conclusion: Galileowas 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 claimholds 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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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 samelaws of physics.

Einstein’s Questions (1907):

Does an “extended” version of this principle hold foraccelerated observers?

How does extending the principle help us to understandgravity?

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 oneanother and are urged by equal accelerative forces along parallellines, they will all continue to move with respect to one another inthe same way as they would if they were not acted on by thoseforces. (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 toinertial 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 beinherent 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 toAcceleration

Either observer canclaim to be at rest,disagree about whetherthere is gravity

(I) and (II) can beaccounted for withgravity or withacceleration

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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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

Gravitational Time Dilation

Einstein (1907)

Strategy

- Consider accelerated observers in special relativity, usereasoning 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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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

Guiding Principles

Equivalence Principle

- Freely falling frame (gravity + inertia) equivalent to inertialframe

- 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 defininginertial 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-inertialmotion? (Why are some states of motion singled out asinertial?)

Newton’s Answer: motion defined with respect to “spaceitself”

Mach and Einstein: motion defined with respect to otherbodies

Dialogue from Einstein (1914)

Two masses, close enough so that they interact. Consider lookingalong the line between them towards the starry night sky.

Mach My masses carry out a motion, which is at least in partcausally determined by the fixed stars. The law by whichmasses in my surroundings move is co-determined by the fixedstars.

Newton The motion of your masses has nothing to do with the heavenof fixed stars; it is rather fully determined by the laws ofmechanics 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 Icannot believe in this gigantic thing that you speak of and callspace. I can neither see something like that nor conceive of it.Or should I think of your space S as a subtle net of bodiesthat the remaining things are all referred to? Then I canimagine a second such net S′ in addition to S, that is movingin an arbitrary manner relative to S (for example, rotating).Do your equations also hold at the same time with respect toS′?

Newton No

Mach But how do the masses know which “space” S, S′, etc., withrespect to which they should move according to yourequations, how do they recognize the space or spaces theyorient themselves with respect to? . . . I will take, for the timebeing, your privileged spaces as an idle fabrication, and staywith my conception, that the sphere of fixed starsco-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 thelaws → 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”)

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 sphereof fixed stars co-determines the mechanical behavior of my testmasses”

Connection with Equivalence Principle

- Equivalence principle breaks down distinction between inertialand 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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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

Euclid’s Elements

Geometry pre-Euclid

- Assortment of acceptedresults, e.g. Pythagoras’stheorem

- How do these results relate toeach other? How does onegive a convincing argument infavor of such results? Whatwould 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

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 thesame 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 theinterior angles in the same direction less than two right angles,then the two straight lines, if produced in infinitum, meet oneanother in that direction in which the angles less than tworight 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 merelyanalytic) possible?

- What is the subject matter of geometry? Why is geometryapplicable 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 theinterior angles in the same direction less than two right angles,then the two straight lines, if produced in infinitum, meet oneanother in that direction in which the angles less than tworight angles are.

5-ONE Simpler, equivalent formulation: Given a line and a point noton the line, there is one line passing through the point parallelto 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 otherpostulates, 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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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

Introduction

Alternatives for Postulate 5

5-ONE Given a line and a point not on the line, there is one linepassing through the point parallel to the given line.

5-NONE Given a line and a point not on the line, there are no linespassing through the point parallel to the given line.

5-MANY Given a line and a point not on the line, there are many linespassing 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 otherpostulates) leads to contradictions, so 5-ONE must be correct.

Construction: assuming 5-NONE, construct triangles with acommon line as base

Results: sum of angles of a triangle > 180◦; circumference6= 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

Pre-1830 (Saccheri et al.)

Study alternatives to findcontradiction

Prove a number of resultsfor “absurd” geometrieswith 5-NONE, 5-MANY

Nineteenth Century

These are fully consistentalternatives to Euclid

5-NONE: sphericalgeometry

5-MANY: hyperbolicgeometry

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-ONEReplace 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 followingproperties?

Pick an arbitrary point.Circles:

- Nearby have C ≈ 2πR- As R increases,C < 2πR

Angles sum to more thanEuclidean case (fortriangles, 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 hyperboloidsurface:

“Extra space”

Circumference > 2πR

Angles sum to less thanEuclidean case (fortriangles, 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 contradictionfollows from 5-NONE or 5-MANY?

Relative Consistency Proof

If Euclidean geometry is consistent, then hyperbolic / sphericalgeometry 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 = 2πRSpherical 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 Relativity2 Newtonian Gravity

Kepler’s Laws

Inertia and Acceleration

Gravity

Conservatives vs. Einstein3 Equivalence Principle

Extending Relativity4 Using the Equivalence Principle

Gravitational Time Dilation

Light Bending: Trajectory and Speed of

Light5 Rejection of Absolute Space6 Euclidean Geometry

Deductive Structure

Fifth Postulate7 non-Euclidean Geometry

Introduction

Spherical Geometry

Hyperbolic Geometry

Summary8 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

Intrinsic vs. Extrinsic

Geometry on a Surface

What does “geometry of figures drawn on surface of asphere” mean?

Intrinsic geometry

- Geometry on the surface; measurements confined to the2-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 asphere” mean?

Intrinsic geometry

- Geometry on the surface; measurements confined to the2-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 asphere” mean?

Intrinsic geometry

- Geometry on the surface; measurements confined to the2-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-dimensionalspace

- Limits of visualization: 3-dimensional surface embedded in4-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-dimensionalspace

- Limits of visualization: 3-dimensional surface embedded in4-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-dimensionalspace

- Limits of visualization: 3-dimensional surface embedded in4-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 constantSpherical 5-NONE positive converge

Hyperbolic 5-MANY negative diverge

Riemannian Geometry

Curvature allowed to vary from point to point; link with geodesicdeviation 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 constantSpherical 5-NONE positive converge

Hyperbolic 5-MANY negative diverge

Riemannian Geometry

Curvature allowed to vary from point to point; link with geodesicdeviation still holds.