curved space
Post on 01-Jun-2018
240 Views
Preview:
TRANSCRIPT
-
8/9/2019 Curved Space
1/88
An Introduction to Curved Spaces
Surjeet Rajendran
Physics 231
-
8/9/2019 Curved Space
2/88
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
-
8/9/2019 Curved Space
3/88
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
Looks a lot like electromagnetism.Is this Lorentz invariant? Why?
Newtonian Gravity :F = GNMm
r2 r
-
8/9/2019 Curved Space
4/88
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
Why not do something similar for gravity?
Looks a lot like electromagnetism.Is this Lorentz invariant? Why?
Newtonian Gravity :F = GNMm
r2 r
-
8/9/2019 Curved Space
5/88
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
Why not do something similar for gravity?
In fact, we can!
Looks a lot like electromagnetism.Is this Lorentz invariant? Why?
Newtonian Gravity :F = GNMm
r2 r
-
8/9/2019 Curved Space
6/88
First Attempt at Relativistic Gravitation
A and field strength F
First Question: Electromagnetism is represented by a vector potential
-
8/9/2019 Curved Space
7/88
First Attempt at Relativistic Gravitation
A and field strength F
First Question: Electromagnetism is represented by a vector potential
How many indices represent gravity?
-
8/9/2019 Curved Space
8/88
First Attempt at Relativistic Gravitation
A and field strength F
First Question: Electromagnetism is represented by a vector potential
How many indices represent gravity?
Why does the electromagnetic vector potential have 1 index?
-
8/9/2019 Curved Space
9/88
First Attempt at Relativistic Gravitation
A and field strength F
First Question: Electromagnetism is represented by a vector potential
How many indices represent gravity?
Why does the electromagnetic vector potential have 1 index?
Electromagnetism couples to a charge current J
J
is a nice four vector. And naturally couples to A
(JA)
-
8/9/2019 Curved Space
10/88
First Attempt at Relativistic Gravitation
A and field strength F
First Question: Electromagnetism is represented by a vector potential
How many indices represent gravity?
Why does the electromagnetic vector potential have 1 index?
Electromagnetism couples to a charge current J
J
is a nice four vector. And naturally couples to A
(JA)
Gravity couples to mass.
-
8/9/2019 Curved Space
11/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor
-
8/9/2019 Curved Space
12/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor
But T
has two indices.
-
8/9/2019 Curved Space
13/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor
But T
has two indices.
Hard to couple a gravitational vector potential Ag
to T
-
8/9/2019 Curved Space
14/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor
But T
has two indices.
Hard to couple a gravitational vector potential Ag
to T
i.e. cant write AgT
-
8/9/2019 Curved Space
15/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor
But T
has two indices.
Hard to couple a gravitational vector potential Ag
to T
i.e. cant write AgT
Two Possibilities
-
8/9/2019 Curved Space
16/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor
But T
has two indices.
Hard to couple a gravitational vector potential Ag
to T
i.e. cant write AgT
Two Possibilities
A scalar potential coupling to T: T
-
8/9/2019 Curved Space
17/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
In special relativity, mass is equivalent to energy and isrepresented by the stress-energy tensor
But T
has two indices.
Hard to couple a gravitational vector potential Ag
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
-
8/9/2019 Curved Space
18/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
Two Possibilities
T hT
-
8/9/2019 Curved Space
19/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
Two Possibilities
T hT
(Nordstrom)
-
8/9/2019 Curved Space
20/88
First Attempt at Relativistic Gravitation
How many indices represent gravity?
Gravity couples to mass.
Two Possibilities
T hT
Exercise: Show that in this theory, light does not couple togravity
(Nordstrom)
-
8/9/2019 Curved Space
21/88
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
-
8/9/2019 Curved Space
22/88
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
Einstein could have pursued this coupling.Done lots of algebra to see what it
might mean
-
8/9/2019 Curved Space
23/88
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
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
-
8/9/2019 Curved Space
24/88
First Attempt at Relativistic Gravitation
Two 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
Picked a deep, intuitive approachthat gave him the right answer
Computation 6= Comprehension
-
8/9/2019 Curved Space
25/88
First Attempt at Relativistic Gravitation
Two 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
Picked a deep, intuitive approachthat gave him the right answer
Computation 6= Comprehension
As we will see later, this turns out tobe the right coupling
-
8/9/2019 Curved Space
26/88
First Attempt at Relativistic Gravitation
Two 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
Picked a deep, intuitive approachthat gave him the right answer
Computation 6= Comprehension
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
-
8/9/2019 Curved Space
27/88
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.
-
8/9/2019 Curved Space
28/88
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.
True in a deeper sense
-
8/9/2019 Curved Space
29/88
Equivalence Principle
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
-
8/9/2019 Curved Space
30/88
Equivalence Principle
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
Observers have to use physical devices to make measurements
-
8/9/2019 Curved Space
31/88
Equivalence Principle
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
Observers have to use physical devices to make measurements
For forces like electromagnetism, we can conceive of physical devices (such asneutral bodies) that do not respond to electromagnetism
-
8/9/2019 Curved Space
32/88
Equivalence Principle
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
Observers have to use physical devices to make measurements
For forces like electromagnetism, we can conceive of physical devices (such asneutral bodies) that do not respond to electromagnetism
This allows us to define inertial observerswho are not affected by
electromagnetic forces and whose motion is thus different from acceleratedobservers who experience electromagnetism
-
8/9/2019 Curved Space
33/88
Equivalence Principle
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
Observers have to use physical devices to make measurements
For forces like electromagnetism, we can conceive of physical devices (such asneutral bodies) that do not respond to electromagnetism
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?
-
8/9/2019 Curved Space
34/88
Equivalence Principle
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
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?
But is it true that allfreely falling observers are equivalent?
-
8/9/2019 Curved Space
35/88
Equivalence Principle
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
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?
But is it true that allfreely falling observers are equivalent?
P1
P2
No
P1, P2are at different distances with respect to thegravitating earth.
They will fall differently.
E l P l
-
8/9/2019 Curved Space
36/88
Equivalence Principle
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
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?
But is it true that allfreely falling observers are equivalent?
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
-
8/9/2019 Curved Space
37/88
Equivalence Principle
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
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?
But is it true that allfreely falling observers are equivalent?
P1
P2 No
What if P1and P2are inside elevators that prevent themfrom looking outside and seeing each other?
E l P l
-
8/9/2019 Curved Space
38/88
Equivalence Principle
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
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?
But is it true that allfreely falling observers are equivalent?
P1
P2 No
What if P1and P2are inside elevators that prevent themfrom looking outside and seeing each other?
If the observers restrict themselves to solely localmeasurements, they cannot know if they are inertial or not
E i l P i i l
-
8/9/2019 Curved Space
39/88
Equivalence Principle
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
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?
But is it true that allfreely falling observers are equivalent?
P1
P2 No
What if P1and P2are inside elevators that prevent themfrom looking outside and seeing each other?
If the observers restrict themselves to solely localmeasurements, they cannot know if they are inertial or not
Exercise: Devise an experiment to show this
E i l P i i l
-
8/9/2019 Curved Space
40/88
Equivalence Principle
P1
P2
What if P1and P2are inside elevators that prevent them
from looking outside and seeing each other?
If the observers restrict themselves to solely localmeasurements, they cannot know if they are inertial or not
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 ?
-
8/9/2019 Curved Space
41/88
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 ?
-
8/9/2019 Curved Space
42/88
P1
P2
Under gravity, freely falling observers (like P2) are
inertial
What then is gravitation?
Wh t th i it ti ?
-
8/9/2019 Curved Space
43/88
P1
P2
Under gravity, freely falling observers (like P2) are
inertial
What about observers like P1who are at rest withrespect to the surface of the earth?
What then is gravitation?
What then is gravitation?
-
8/9/2019 Curved Space
44/88
P1
P2
Under gravity, freely falling observers (like P2) are
inertial
P1observes gravity (he can just drop a coin and see itfall)
What about observers like P1who are at rest withrespect to the surface of the earth?
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.
What then is gravitation?
What then is gravitation?
-
8/9/2019 Curved Space
45/88
P1
P2
Under gravity, freely falling observers (like P2) are
inertial
P1observes gravity (he can just drop a coin and see itfall)
What about observers like P1who are at rest withrespect to the surface of the earth?
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.
What then is gravitation?
In fact, local observations made by P1are in-distinguishable fromthat of an accelerated platform in Minkowski space
What then is gravitation?
-
8/9/2019 Curved Space
46/88
P1
P2
Under gravity, freely falling observers (like P2) are
inertial
P1observes gravity (he can just drop a coin and see itfall)
What about observers like P1who are at rest withrespect to the surface of the earth?
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.
What then is gravitation?
In fact, local observations made by P1are in-distinguishable fromthat of an accelerated platform in Minkowski space
Exercise: Show this in a simple setup
What then is gravitation?
-
8/9/2019 Curved Space
47/88
P1
P2
Under gravity, freely falling observers (like P2) are
inertial
P1observes gravity (he can just drop a coin and see itfall)
What about observers like P1who are at rest withrespect to the surface of the earth?
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.
What then is gravitation?
In fact, local observations made by P1are in-distinguishable fromthat of an accelerated platform in Minkowski space
Einstein: This is not just true for particle mechanics, but
true for all forces of nature
Consequences
-
8/9/2019 Curved Space
48/88
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
-
8/9/2019 Curved Space
49/88
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
-
8/9/2019 Curved Space
50/88
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
-
8/9/2019 Curved Space
51/88
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
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
-
8/9/2019 Curved Space
52/88
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
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)
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
-
8/9/2019 Curved Space
53/88
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
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)
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?
-
8/9/2019 Curved Space
54/88
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?
-
8/9/2019 Curved Space
55/88
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
-
8/9/2019 Curved Space
56/88
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
-
8/9/2019 Curved Space
57/88
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?
-
8/9/2019 Curved Space
58/88
How do we make these quantitative?
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
How do we make these quantitative?
-
8/9/2019 Curved Space
59/88
How do we make these quantitative?
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
-
8/9/2019 Curved Space
60/88
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?
-
8/9/2019 Curved Space
61/88
What do we want out of curved spaces?
-
8/9/2019 Curved Space
62/88
p
We want to be able to describe objects like spheres, torii, Moebius strips...
What do we want out of curved spaces?
-
8/9/2019 Curved Space
63/88
p
We want to be able to describe objects like spheres, torii, Moebius strips...
There is a sense in which these objects have a dimensionality about them (forexample, all the objects above seem to be 2 dimensional)
What do we want out of curved spaces?
-
8/9/2019 Curved Space
64/88
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
We want to be able to describe objects like spheres, torii, Moebius strips...
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)
What do we want out of curved spaces?
-
8/9/2019 Curved Space
65/88
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
What do we want out of curved spaces?
-
8/9/2019 Curved Space
66/88
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
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.
What do we want out of curved spaces?
-
8/9/2019 Curved Space
67/88
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
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)
What do we want out of curved spaces?
-
8/9/2019 Curved Space
68/88
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
What do we want out of curved spaces?
-
8/9/2019 Curved Space
69/88
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
What do we want out of curved spaces?
-
8/9/2019 Curved Space
70/88
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
What do we want out of curved spaces?
-
8/9/2019 Curved Space
71/88
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
-
8/9/2019 Curved Space
72/88
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
-
8/9/2019 Curved Space
73/88
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
-
8/9/2019 Curved Space
74/88
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",!"} 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
-
8/9/2019 Curved Space
75/88
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",!"} 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
-
8/9/2019 Curved Space
76/88
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
-
8/9/2019 Curved Space
77/88
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
-
8/9/2019 Curved Space
78/88
Which of these are manifolds?
Take R2. Identify edges along the directions indicatedby the arrows and get new spaces.
(a) (b) (c) (d)
-
8/9/2019 Curved Space
79/88
Examples
-
8/9/2019 Curved Space
80/88
Which of these are manifolds?
Take R2. Identify edges along the directions indicatedby the arrows and get new spaces.
(a) (b) (c) (d)
cylinder mobius strip
Examples
-
8/9/2019 Curved Space
81/88
Which of these are manifolds?
Take R2. Identify edges along the directions indicatedby the arrows and get new spaces.
(a) (b) (c) (d)
cylinder mobius strip torus
Examples
-
8/9/2019 Curved Space
82/88
Which of these are manifolds?
Take R2. Identify edges along the directions indicatedby the arrows and get new spaces.
(a) (b) (c) (d)
cylinder mobius strip torus klein bottle
More Examples
-
8/9/2019 Curved Space
83/88
Which of these are manifolds?
(a) (b) (c)
More Examples
-
8/9/2019 Curved Space
84/88
Which of these are manifolds?
(a) (b) (c)
A chart that coversthe point of
intersection doesnot map into open
sets
More Examples
-
8/9/2019 Curved Space
85/88
Which of these are manifolds?
(a) (b) (c)
A chart that coversthe point of
intersection doesnot map into open
sets
The line is R1andthe plane is R2. Sothere is no way wecan have a cover.
More Examples
-
8/9/2019 Curved Space
86/88
Which of these are manifolds?
(a) (b) (c)
A chart that coversthe point of
intersection doesnot map into open
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
-
8/9/2019 Curved Space
87/88
-
8/9/2019 Curved Space
88/88
Next Class: Vectors, Tensors and MetricSpaces
(chapter 2 of Carroll)
top related