Curved Space

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An Introduction to Curved Spaces

Surjeet Rajendran

Physics 231


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Why Curved Space?

Special Relativity beautifully describes electromagnetism

F= J

[F] = 0

The theory is Lorentz invariant, preserves the fact thatsignals do not travel faster than the speed of light


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Why Curved Space?


F= J

[F] = 0


Looks a lot like electromagnetism.Is this Lorentz invariant? Why?

Newtonian Gravity :F = GNMm

r2 r


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Why Curved Space?


F= J

[F] = 0


Why not do something similar for gravity?



r2 r


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Why Curved Space?


F= J

[F] = 0


Why not do something similar for gravity?

In fact, we can!



r2 r


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First Attempt at Relativistic Gravitation

A and field strength F

First Question: Electromagnetism is represented by a vector potential


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How many indices represent gravity?


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Why does the electromagnetic vector potential have 1 index?


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Electromagnetism couples to a charge current J

J

is a nice four vector. And naturally couples to A

(JA)


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Electromagnetism couples to a charge current J

J

is a nice four vector. And naturally couples to A

(JA)

Gravity couples to mass.


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In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor


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But T

has two indices.


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But T

has two indices.

Hard to couple a gravitational vector potential Ag

to T


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But T

has two indices.


to T

i.e. cant write AgT


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But T

has two indices.


to T

i.e. cant write AgT

Two Possibilities


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But T

has two indices.


to T

i.e. cant write AgT

Two Possibilities

A scalar potential coupling to T: T


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But T

has two indices.


to T

i.e. cant write AgT

Two Possibilities

A scalar potential coupling to T: T

A symmetric tensor potential h coupling to T

: hT


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Two Possibilities

T hT


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Two Possibilities

T hT

(Nordstrom)


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Two Possibilities

T hT

Exercise: Show that in this theory, light does not couple togravity

(Nordstrom)


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First Attempt at Relativistic GravitationTwo Possibilities

T hT

Gravity does not couple to light,eventhough it is a form of energy

(Nordstrom)

Einstein did not like this kindof discrimination. Feltgravitation

should be universal


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T hT


(Nordstrom)


should be universal

Einstein could have pursued this coupling.Done lots of algebra to see what it

might mean


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T hT


(Nordstrom)


should be universal

Einstein could have pursued this coupling.Done lots of algebra to see what it

might mean

Instead, picked a deep, intuitive approachthat gave him the right answer


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Two Possibilities

T

hT


(Nordstrom)


should be universal

Picked a deep, intuitive approachthat gave him the right answer

Computation 6= Comprehension


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Two Possibilities

T

hT


(Nordstrom)


should be universal



As we will see later, this turns out tobe the right coupling


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Two Possibilities

T

hT


(Nordstrom)


should be universal



As we will see later, this turns out tobe the right coupling

Relativistic gravitation can be obtained purely from flat space-time notions. Its

physical effects are easier to grasp with Einsteins curved space view-point


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Equivalence Principle

Everything falls at the same rate under gravity,irrespective of its mass or composition.

This is strange....

but

Every body perseveres in its stateof being at rest or of moving

uniformly straight forward exceptinsofar as it is being compelled to

change its state by forcesimpressed.

Maybe things are just all moving along in straight lines.

But, straight lines in a curved space.


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Everything falls at the same rate under gravity,irrespective of its mass or composition.

This is strange....

but

Every body perseveres in its stateof being at rest or of moving

uniformly straight forward exceptinsofar as it is being compelled to

change its state by forcesimpressed.

Maybe things are just all moving along in straight lines.

But, straight lines in a curved space.

True in a deeper sense


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One of the deep concepts from special relativity is that we should not rely on

abstract philosophical notions but rather ask what observers can physicallymeasure


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Observers have to use physical devices to make measurements


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For forces like electromagnetism, we can conceive of physical devices (such asneutral bodies) that do not respond to electromagnetism


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This allows us to define inertial observerswho are not affected by

electromagnetic forces and whose motion is thus different from acceleratedobservers who experience electromagnetism


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One of the deep concepts from special relativity is that we should not rely onabstract philosophical notions but rather ask what observers can physically

measure



This allows us to define inertial observerswho are not affected by

electromagnetic forces and whose motion is thus different from acceleratedobservers who experience electromagnetism

But, if everything falls the same way under gravity, how can we even separatean inertial observer from an observer who is freely falling under gravity?


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measure


But is it true that allfreely falling observers are equivalent?


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measure



P1

P2

No

P1, P2are at different distances with respect to thegravitating earth.

They will fall differently.

E l P l


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measure



P1

P2

No

P1, P2are at different distances with respect to thegravitating earth.

They will fall differently.

So P1can look at P2and realize that he is falling under gravity

E l P l


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measure



P1

P2 No

What if P1and P2are inside elevators that prevent themfrom looking outside and seeing each other?

E l P l


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measure



P1

P2 No


If the observers restrict themselves to solely localmeasurements, they cannot know if they are inertial or not

E i l P i i l


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measure



P1

P2 No



Exercise: Devise an experiment to show this

E i l P i i l


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P1

P2

What if P1and P2are inside elevators that prevent them

from looking outside and seeing each other?


This is certainly true for particle mechanics. But what about other forces ofnature like electromagnetism, weak and strong nuclear forces? Or even the

forces caused by the recently discovered Higgs boson?

Einsteins AssertionThe laws of nature are such that a freely falling observerin a gravitational field

who only relies on localmeasurements cannot know that he is around agravitational field. His local observations will be those of an inertial observer

in Minkowski space.

Wh h i i i ?


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What then is gravitation?

P1 P2

Einsteins Assertion

The laws of nature are such that a freely fallingobserverin a gravitational field who only relies on

localmeasurements cannot know that he is around agravitational field. His local observations will be those

of an inertial observer in Minkowski space.

What if P2is very close to P1?

P1can recognize gravity by making measurements

sufficiently close to himThese measurements can depart from those of inertial

observers in Minkowski space

Gravitation is then the deviations from Minkowski space thatthe freely falling observer can locallymeasure

Wh t th i it ti ?


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P1

P2

Under gravity, freely falling observers (like P2) are

inertial


Wh t th i it ti ?


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P1

P2


inertial

What about observers like P1who are at rest withrespect to the surface of the earth?




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P1

P2


inertial

P1observes gravity (he can just drop a coin and see itfall)


But, even though P1is at rest, he is acted upon by non-gravitational

forces in order to cancel the gravitational force on him from the Earth.




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P1

P2


inertial






In fact, local observations made by P1are in-distinguishable fromthat of an accelerated platform in Minkowski space



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P1

P2


inertial







Exercise: Show this in a simple setup



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P1

P2


inertial



But, even though P1is at rest, he is acted upon by non-gravitationalforces in order to cancel the gravitational force on him from the Earth.



Einstein: This is not just true for particle mechanics, but

true for all forces of nature

Consequences


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P1

P2

If everything falls the same way under gravity, much like

any other free falling observer (P2) light will also fallrelative to an observer (P1) on the ground

Consequences

Consequences


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P1

P2



Consequences

Consequences


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P1

P2



Consequences

Consequences


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P1

P2



Consequences

P3

Observer P3is also at rest with respect to the surfaceof the earth (and observer P1). These are accelerated

observers. Hence, local measurements such as theticking rates of their clocks will not be equal

(consequence of special relativity)

Consequences


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P1

P2



Consequences

P3




If P1and P3send light signals back and forth, that ismuch like accelerated observers sending signals back

and forth. The light signals will be doppler shifted.

Consequences


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P1

P2



Consequences

P3




If P1and P3send light signals back and forth, that ismuch like accelerated observers sending signals back

and forth. The light signals will be doppler shifted.

Exercise: Compute this Doppler shift. Is it compatible with

elementary notions of flat space?

Are these principles true?


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Are these principles true?Freely falling observers are inertial.

They cannot do a local measurement to realize that they

are in a gravitational field

An observer who is at rest on the surface of the earth isan accelerated observer

Are these principles true?


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Are these principles true?Freely falling observers are inertial.

They cannot do a local measurement to realize that they

are in a gravitational field

An observer who is at rest on the surface of the earth isan accelerated observer

Test CaseTake a charged particle.

Let it freely fall. It is inertial.

Does it radiate?

Place a charged particle on the surface of the Earth. It is accelerated.Does it radiate?

e-e-

Summary


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Summary

In special relativity, inertial observers were observerson whom there were no external forces

Their inertial nature was global. One

observer could look at a distant inertialobserver and this observation will notmaking him doubt his inertial nature.

t

x

t

x

t

x

Minkowski space was invariant under globalLorentztransformations and spatial translations

Summary


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SummaryMinkowski space was invariant under globalLorentz

transformations and spatial translations

P1

P2

The presence of the earth breaksthis globalsymmetry.

Since all observers feel gravity, this breaks theseglobal symmetries for all observers

General Relativity (Relativistic Gravitation)

Local measurements of freely falling observers areequivalent to those of inertial observers in

Minkowski space

Gravity appears to a local observer as a deviationfrom Minkowski space

These properties imply that gravity must curve

space-time (doppler shift of light)

How do we make these quantitative?


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In a small, local region around them, freely falling observers

experience Minkowski (or flat) space-time

Gravitation is represented by deviations from Minkowski(or flatness) in this local region

Further, gravity must curve space-time



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In a small, local region around them, freely falling observers

experience Minkowski (or flat) space-time

Gravitation is represented by deviations from Minkowski(or flatness) in this local region

Further, gravity must curve space-time

This is a lot like...

the flatness of the Earth!P

Near the point P, the Earth looks very flat. The fact that it is

curved shows up in small deviations of the flat geometry

The Geometry of Curved Spaces


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The Geometry of Curved Spaces

PP

Near the point P, the Earth looks very flat.

The fact that it is curved shows up in small deviations offlat geometry

Questions

How do we precisely define the fact that near the point Pthe earth is very flat? (Manifolds)

How do our favorite notions from flat space such as vectorsand tensors translate to curved space?

What quantities capture the deviation from flat space at thepoint P? (this after all is gravity - and the answer is

curvature)

How can different observers compare their measurements?

What do they agree on? What are the invariants?


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What do we want out of curved spaces?


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p

We want to be able to describe objects like spheres, torii, Moebius strips...



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p


There is a sense in which these objects have a dimensionality about them (forexample, all the objects above seem to be 2 dimensional)



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p

Near any point P of the space, there should be a small enough region thatlooks like it is flat space. In fact, the dimensionality of this flat space is tiedto the notion of the spaces dimensionality. For all the examples above,

small enough regions look like R2


P

P P

There is a sense in which these objects have a dimensionality about them (forexample, all the objects above seem to be 2 dimensional)



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p

We could of course describe them based on their embeddingin flat space.e.g. sphere is the set of points in R3such that x2+ y2+ z2= C

P

P P



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p


P

P P

But this is cumbersome - these spaces have a life of their own independentof how we embed them in a higher dimensional space. We want to be ableto describe those properties without worrying about the embedding or

how the higher dimensional space is parametrized.



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p


P

P P

But this is cumbersome - these spaces have a life of their own independentof how we embed them in a higher dimensional space. We want to be ableto describe those properties without worrying about the embedding or

how the higher dimensional space is parametrized.

Further, as we will see, there are examples of 2d surfaces that cannot beembedded in R3(but can be embedded in R4). But they will have many other

properties shared by other 2d surfaces (rather than 3d spaces that are

more naturally embedded in R4)



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p

Want to describe these spaces based purely on their intrinsicpropertiesrather than their extrinsic embedding. After all, we dont describe R2on thebasis of how it fits into R3, but rather on its own merits. We should treat

these spaces in the same way.

P

P P



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p

Want to describe these spaces based purely on their intrinsicpropertiesrather than their extrinsic embedding. After all, we dont describe R2on thebasis of how it fits into R3, but rather on its own merits. We should treat

these spaces in the same way.

P

P P

In Rnwe know what it means for one point to be near another, based onthe Euclidean distance between points. But if we are not to make use of theembedding, how do we intrinsically define the notion of one point (Q) being

near P so that P agrees that the geometry between P and Q is flat, unlikesay that point Q that is far from P and is hence allowed to be curved?

Q

Q QQ

Q Q



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p

One we have an intrinsic description where we can define how close onepoint is to another, we would like to understand how to extend our usualnotions of continuity and differentiability for functions on these spaces.

P

P P

Q

Q QQ

Q Q



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p

One we have an intrinsic description where we can define how close onepoint is to another, we would like to understand how to extend our usualnotions of continuity and differentiability for functions on these spaces.

P

P P

With that knowledge, we can talk about derivatives of such functions at alocal point P. These derivatives are linear and we will show that these

describe a vector space at P. This vector space will be used to generalize ournotions of vectors and tensors from flat space.

Q

Q QQ

Q Q

Manifold


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U1,1U2,2

U3,3

U4,4

The Set M

A chartor coordinatesystem consists of a subset U of M

along with a one to onemap !: U-> Rn such that that

image !(U) is open in Rn

A set V is open in Rnif for any point x in V there is some r

so that any point y satisfying |x - y| < r is also in V

Manifold


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U1,1U2,2

U3,3

U4,4

The Set MA chartor coordinatesystem consists of a subset U of M along with a one to

onemap !: U-> Rn such that that image !(U) is open in Rn

An atlasis a collection of charts {U",!"} so that

1. The union of U"is equal to M; i.e. the U"coverM.

2. The charts are woven smoothly together; i.e. if two charts overlap

1

:(U U) Rn

(U U) Rn

is onto and continuous in Rn

Manifold


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U1,1U2,2

U3,3

U4,4



An atlasis a collection of charts {U",!"} that coverM and the charts are

woven together smoothly where they intersect

A manifoldis simply a set M along with a maximal atlas, i.e. an atlas thatcontains every possible such chart

Manifold


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U1,1U2,2

U3,3

U4,4



An atlasis a collection of charts {U",!"} that coverM and the charts are

woven together smoothly where they intersect

A manifoldis simply a set M along with a maximal atlas, i.e. an atlas thatcontains every possible such chart

Basically, we use Rnto inducea topology on M

Manifold


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U1,1U2,2

U3,3

U4,4

Manifold Construction forPhysicists

Take the space M - for any point on the space find a one-to-one map that

takes points around that space into open sets in Rn

Find a set of maps that cover every point in the space. Make sure that wherethey overlap, they overlap nicely

Manifold


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U1,1U2,2

U3,3

U4,4

Manifold Construction forPhysicists

Take the space M - for any point on the space find a one-to-one map that

takes points around that space into open sets in Rn

Find a set of maps that cover every point in the space. Make sure that wherethey overlap, they overlap nicely

How do these notions fix the dimensionality of the space?

Examples


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Which of these are manifolds?

Take R2. Identify edges along the directions indicatedby the arrows and get new spaces.

(a) (b) (c) (d)


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Examples


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(a) (b) (c) (d)

cylinder mobius strip

Examples


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(a) (b) (c) (d)

cylinder mobius strip torus

Examples


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(a) (b) (c) (d)

cylinder mobius strip torus klein bottle

More Examples


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(a) (b) (c)

More Examples


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(a) (b) (c)

A chart that coversthe point of

intersection doesnot map into open

sets

More Examples


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(a) (b) (c)



sets

The line is R1andthe plane is R2. Sothere is no way wecan have a cover.

More Examples


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(a) (b) (c)



sets

The line is R1andthe plane is R2. Sothere is no way wecan have a cover.

Smoothconstruction from a

plane. This is amanifold called the

Real Projective

Space RP2


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Next Class: Vectors, Tensors and MetricSpaces

(chapter 2 of Carroll)

Curved Space

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