Download Lecture 8: Curved Spaces

EPGY Summer Institute
Special and General Relativity 2012
Lecture 8: Curved Spaces
With the necessity of curved geodesics within regions with significant energy or mass concentrations we need to
understand curved spacetime. However, since we have trouble even visualizing and understanding flat spacetime, we
will will approach curved spacetimes slowly. We begin by studying ordinary two-dimensional curved spaces since we
can visualize and construct models of these. We will introduce the tools and methods to explore curvature in these
simpler examples then use them to analyze higher dimensional curved spaces and spacetime.
Recall Euclid’s 5th postulate, most simply stated as parallel lines never meet. In the 1800’s serious consideration
was given to spaces where this postulate does not hold. Mathematicians such as, Gauss, Riemann, Lobachevskii
formulated the field of non-Euclidean geometry.
Let’s begin by examining the subspace R2 (the flat infinite plane) embedded into R3 .
Or, to make a more concrete example, consider the flat unit square, I 2 , embedded in R3 .1 A square piece of
paper laid flat is an example of such a space. Euclidean geometry states that a triangle inscribed on this plane will
have the internal angles summing to π radians or 180o. The sides of this triangle need to be geodesics –straight lines
in I 2 .
Let’s now consider our first curved space, S 2 , the 2-sphere, an example of which can be a basketball or the surface
of the Earth (approximately). Note this 2-sphere has no boundary (∂S 2 = ∅ = ∂∂B 3 ). It is generally true that
∂ 2 B i = ∅.)2 . Thus we see that the topology of S 2 is not the same as R2 (being closed and bounded). What is
important for us is to analyze the geometry of S 2 . The first thing to note is that if we draw a triangle in S 2 , where
each side is a geodesic, the interior angles will no longer addP
up to π. For example, we can create a triangle on the
surface of the Earth that is comprised of three right angles, i θi = 32 π.
The sum of the interior angles will depend on the size of the triangle drawn. For example, go out into your
backyard and draw a triangle on the ground, the triangles will add up to π. This is because your triangle is tiny
compared to the Earth. Recall this is just a result of S 2 being a differentiable manifold – on small scales it appears
as R2 . The fact that the interior angles add up to greater than π is that S 2 is a curved space.
Gaussian curvature
For two-dimensional surfaces we define3 the Gaussian curvature, K at a point p as the inverse of the
product of the two radii of curvature for perpendicular geodesics passing through p.
p
1
K =
R1 R2
R1
R2
For the 2-sphere we have that R1 = R2 no matter where you are on the surface, thus K = R12 . Since this value
is the same for all points p on the surface, S 2 is a space of constant positive curvature. Why positive? Well,
there’s another two dimensional surface that can be defined as a space of constant negative curvature, H 2 . This is a
1 A brief word on mathematical notation of spaces (sets). R is the set of reals, or the one-dimensional Euclidean space, Rn is ndimensional Euclidean space. S n is the n-sphere, the locus of points equidistant from a point in Rn+1 . B n the n-ball, the set of points
that lie on S n−1 and within. Note, the boundary of B n+1 (labeled ∂B n+1 ) is S n . And ∂S n = ∅ (n-spheres have no boundary). Other
spaces will be introduced as we proceed.
2 In plain words: The boundary of a space never has a boundary itself. Or, the boundary of a boundary is the empty set.
3 This definition is a little too simplistic, however it will suit our needs for now.
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space where at all points p ∈ H 2 the radii of curvature of two perpendicular curves that pass through p are the same
but lie on the opposite sides of the surface. In this case K = − R12 .
R1
p
R2
Locally, we can draw this as a saddle shape. A triangle inscribed in the center will have interior angles adding
up to less than π. So a classification of curvature for 2-d surfaces can be defined as follows (where θi are the interior
angles of a triangle),
K >0
K =0
K <0
3
X
:
i=1
3
X
:
i=1
3
X
:
θi > π
θi = π
θi < π
i=1
So one method to determine if a space is curved is to just draw triangles and add up the interior angles. Note
that the saddle shaped surface just discussed is not the space of constant negative curvature (H 2 is sometimes called
the Lobachevskii plane.), for as you move away from the center it becomes positively curved. It turns out you can
not embed H 2 into R3 (you actually need R6 to do so faithfully). So we have no hope of visualizing such a space
(but they are important nonetheless).
Other properties distinguish these three cases. In terms of Euclid’s 5th postulate the following holds for spaces
of constant curvature.
K = 0 Through any point not on a line, there is exactly one line that is parallel, and never intersects, the first line.
K > 0 Through any point not on a geodesic line, all geodesics through that point intersect the first line.
K < 0 Through any point not on a geodesic line, there are an infinite number of geodesic lines that do not intersect
the first line.
The curvature can vary from point to point on a surface and the value at a point can be obtained from the
metric. This was Gauss’s Theorema Egregium (remarkable theorem) where the following formula gives the Gaussian
curvature as a function of the metric.
"
(
(
2 )
2 )#
1
1
∂ 2 g11
∂ 2 g22
1
∂g11 ∂g22
∂g11 ∂g22
∂g11
∂g22
K =
+
−
−
+
+
+
2g11 g22
∂(x2 )2
∂(x1 )2
2g11 ∂x1 ∂x1
∂x2
2g22 ∂x2 ∂x2
∂x1
Don’t worry, you are not expected to use or remember this expression. But recall that the metric is the coefficients
in our metric equation (for coordinates x1 and x2 ),
dl2
=
g11 dx1 dx1 + g22 dx2 dx2 + g12 dx1 dx2 + g21 dx2 dx1
Where, for R2 we have g11 = g22 = 1, & g12 = g21 = 0.
We want to explore an easier method to calculate the curvature that we will then use to explore higher dimensional
spaces and spacetime.
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Parallel transport of vectors
To classify the amount of curvature in a region we introduce a method that is actually used in the full theory of
relativity. Again, we will begin with simple surfaces and then show how to do it in spacetime.
Consider the following exercise first considered in R2 .
1) Draw a closed polygon with sides comprised of geodesics, that is lines of shortest distance. The vertices will
be labelled a1 , a2 , ...an . The length of each line is arbitrary for now and the vertices should be numbered in a
counterclockwise manner around the polygon.
2) Starting from vertex a1 draw a vector in any direction. Note the angle between this vector and the first line
segment l1−2 , call it θ1−2 .
3) Slide the vector along this line segment such that the angle between the vector and the line is always θ1−2 until
the next vertex a2 is reached. This operation is called parallel transport.
4) Note the angle between the vector and the next line segment, θ2−3 and now parallel transport the vector along
this line segment.
5) Repeat all the way around the polygon until you have returned to a1 .
6) Note the relationship between the initial vector and the parallel transported angle. This angle is called the
deficit angle, ǫ.
It will be seen that this deficit angle can be used as a measure of the net curvature in the surface bounded by the
polygon. In more advanced cases an infinitesimal rectangle is chosen so as to measure the curvature at a point. Let’s
see how the relation is to be found through examples.
Examples of two dimensional surfaces
The flat plane R2
You can verify for yourself that drawing any polygon on a piece of paper and parallel transporting a vector around
it, the vector returns pointing in the same direction. (You can use a pencil and maintain the angle between the
pencil and the lines as you go around). This tells us there is no curvature in the flat plane.
The 2-sphere, S 2
Here is where it gets interesting. To find the relation, we will choose a large triangle as in the previous example
–the triangle comprised of three interior right angles. We will begin at the ‘equator’ and break it into three legs.
ǫ=
θ1−2 = 0o
θ2−3 = 90o
π
2
θ3−1 = 180o
We see that the deficit angle is π2 in the counterclockwise direction, this is because the surface is curved within
the triangle. You can easily verify this result with a pencil and a basketball. How do we obtain the curvature in this
case? Note that the area enclosed by the triangle is 81 of the total area of the surface (the two lower vertices are 90o
in ‘longitude’).
The net Gaussian curvature within the triangle can be found by the following relation,
ǫ
=
KA
where A is the area bounded by the polygon. Thus we have,
ǫ=
π
2
=
K
=
1
π
K (4πR2 ) = KR2
8
2
1
R2
3
the result we stated before. Since the curvature is the same at all points the result will be the same no matter how
small we make the polygon.
A cone
As a fun little exercise we can check the curvature of a cone. How do you make a cone? Take a piece of paper
(or I 2 ) and cut a wedge out. Identify the two edges of the wedge and you have a cone (tape together the two sides).
Let’s examine the curvature properties of this cone by examining the plane with a wedge removed. Consider two
paths, one which encloses the apex and one which does not.
ǫ=0→K=0
ǫ
Notice that in the first case there is no curvature within the square while in the second case there is. Remember
that the two green dots are identified – they are the same spot. To see the deficit angle, notice the angle the vector
makes with the line of the wedge, it is different and the result is a counterclockwise angle. Thus there is a net positive
curvature within that triangle. You should convince yourself that a deficit angle will be non-zero as long as the curve
encloses the apex (black dot). Thus all of the curvature of a cone is concentrated at its apex, the other parts have
no curvature.
A negatively curved cone
You can create the negatively curved variant of a cone by reversing the process above. If, instead of cutting
a wedge out, you add a wedge to the flat plane, you will get a negatively curved surface where the curvature is
concentrated at the apex. You can do this yourself by cutting along a line to the center of the sheet. From another
piece of paper cut a wedge that can be squeezed in between the two edges of the slit. You can see the result is our
saddle shaped surface. If you then parallel transport a vector around the apex you should find the deficit angle to
be in the opposite direction of the traversed path (e.g. rotated clockwise if went around loop in a ccw way).
A cylinder
It will be left for you as an exercise to see that a cylinder has no curvature. This may seem like an odd result but is
true. The cylinder has no intrinsic curvature, that means no curvature can be measured by small local measurements.
However it does have extrinsic curvature which is seen just by embedding it in R3 . The cylinder differs in global
properties, that is, it has different topology than R2 but the local geometry is the same.
Higher dimensional spaces
Now that we have the conceptual tools to find curvature in two dimensional spaces let’s have a quick look at curved
higher dimensional spaces. It turns out the quantification of curvature in higher dimensional spaces is more complex
than in 2-dim as there are many planes along which the space can curve. By drawing a small rectangle on 2-dim
surfaces there is only one orientation possible. However in three dimensions there are 6 components required to
specify the curvature, in spacetime there are 20.4
Thus we see the method above needs to be generalized to account for these other components. The idea however
is the same. If we parallel transport a vector V µ around a parallelogram spanned by geodesics and the change is
dV µ , the analog of ǫ = KA can be represented as follows,
dV µ
=
µ
Rνλρ
V ν Aλρ
µ
where Aλρ is related to the area enclosed by the parallelogram and Rνλρ
is the Riemann curvature tensor. This
tensor quantifies the curvature in n-dimensional space, it is the central object of study. It can be expressed as a
(somewhat complex) function of the metric g µν . Again, the main point being that the metric describes the geometry,
which includes the curvature at all points in space. This expression is also valid for spacetime. Thus we have a
quantity that will help us describe the curvature of spacetime.
4 In
general there are Nn =
1 2 2
n (n
12
− 1) independent components of the curvature.
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Einstein equation
In the years 1907 until 1915 Albert Einstein was learning about non-Euclidean geometry and trying to understand
how to relate the Riemann curvature tensor to the distribution of energy and momentum. The later quantity was
understood before to be the Stress-Energy tensor, the problem was how to relate it to the Riemann curvature.
Various forms were attempted and discarded. He understood that there should be an equation relating a quantity
that is a function of the Riemann tensor to the stress-energy tensor but took quite a while to find the correct relation.
A modern study of this relation relies upon known transformation properties of the various tensors to reduce it to
the correct form. Since we do not have the time or mathematical machinery to tackle this relation we will give a
qualitative statement and show its form. Some connections to what we have covered previously will be made.
The Einstein field equation can be described in words in the following way,
Quantity representing the curvature
of spacetime at a particular point
=
Quantity representing energy and momentum
[Universal constant] ×
density at a point in spacetime.
The names of these quantities and the mathematical expression are,
Stress-Energy tensor
8πG
[Einstein tensor] =
c4
(Energy-momentum tensor)
8πG
Gµν =
T µν
c4
−43 m
Note that 8πG
c4 = 2.07 × 10
J testify to the weak nature of gravity (or put another way it takes a lot of energy to
curve spacetime). The Einstein tensor is usually expressed in the form,
Gµν
1
= Rµν − g µν R
2
P
µ
. The
where Rµν is the Ricci tensor it is related to the full Riemann curvaturePtensor as Rµν = σ,λ,ρ g σν g λρ Rσλρ
µν
other term, R is the Ricci scalar, obtained from the Ricci tensor as R = µν gµν R .
Of course you are not expected to understand all of this but the point to take away is the following, The left
side of the Einstein equation is related to the Riemann curvature tensor which is obtained by parallel transporting a
vector around a parallelogram just like for two dimensional surfaces.
Lastly a few words about the energy-momentum tensor. For simple systems (e.g. a system of dust particles) this
tensor is directly related to the energy-momentum 4-vector introduced a few lectures back. Recall that q µ = cpµ , it
is just the 4-momentum times c. The stress energy tensor can be expressed then as T µν = q µ ⊗ q ν where ⊗ is an
outer product (input two vectors, returns a matrix). Again, familiarity is not expected here but the connections to
our earlier description of energy and momentum are evident.
What this equation states is that given some distribution of matter and energy (T µν ) the geometric properties of
spacetime (g µν from Gµν ) is determined. From Gµν we can find the metric, which will tell us the timelike geodesics
along which free particles travel. From the metric we also find the lightlike geodesics along which light will follow.
The Einstein equation is non-linear
This equation is a very difficult equation to solve (find the metric given the energy distribution). When Einstein
introduced it, he thought it would be many years before anyone would find a solution to it. However, in the same
year Karl Schwarzschild found a solution for the spherically symmetric case. (This will be discussed in the next
lecture). There is no general solution to the equation, but over the years many specific solutions have been found
for simplified conditions.
The question is, why is this equation so difficult to solve? One reason why this is so difficult is that the equation
is nonlinear. This means that the equation involves many powers of the variables, cross terms, including several
derivatives of the variables (metric). To compare, consider the force equation for electric forces, F = qE. This
equation is linear in the field E. Consider two problems, 1 and 2, with two separate solutions, F1 and F2 . If we
consider the combined problem, where there are three charges now. To solve this problem we simply add the solutions
for the two separate problems,
~ 2 = q+ (E
~1 + E
~ 2 ).
~ 1 + q+ E
F~12 = F~1 + F~2 = q+ E
You may recall that this process is called linear superposition. For non-linear equations, we can not break down the
problem like this. The solution to the combined problem is not the same as the sum of the first two. This can easily be
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demonstrated by postulating that the electric force is no longer linear but obeys F = qE 2 . For now, F1 = qE12 , F2 =
qE22 , but the sum of these two is not equal to the combined problem, F12 = q(E1 + E2 )2 6= qE12 + qE22 = F1 + F2 . The
other difficulty with finding solutions is that the gravitational field couples to itself. In Maxwells theory we discussed
the energy content of electric and magnetic fields, U ∼ E 2 or B 2 . The same holds true for gravitational fields.
According to SR, mass and energy are related so that the field that is being mediated is its own source, creating
more field etc. The additional content is small however. In our modern viewpoint we would say that curvature of
space adds energy, which then curves space.
Even though the theory is very difficult to deal with, it has passed every experimental test so far. Beginning
with Eddingtons expeditions, to flying clocks about the world, to measuring redshift of light, etc. etc. The Gravity
Probe B, launched in 2003, provided one of the strictest tests of the theory to date. However since the tie between
energy density and curvature is so weak, it is very difficult to do experimental verifications. Luckily, we have many
very massive objects in the universe which we can observe and test the results.
More Formal Statement of General Relativity
For those of you how are more mathematically inclined, a more formal definition of general relativity is presented.
I Spacetime is represented by a four dimensional differentiable pseudo-Riemannian manifold endowed
with a symmetric (pseudo-Riemannian) metric (tensor) g µν .
1) g µν is nonsingular (doesnt become infinite anywhere) and its spatial terms are negative and its
temporal term is positive (i.e. signature is (+ − −−)). (The metric has some other, more subtle
properties).
2) g µν is determined by the Einstein field equation,
Gµν
=
8πG µν
T
c4
where T µν is a tensor describing the energy and momentum density in some region. Gµν is a
µ
related to the curvature (through Rσλρ
) in this region and is a function of the metric.
II There exists privileged classes of curves in the manifold.
1) Ideal clocks travel along timelike curves and measure the parameter τ (the proper time).
2) Free particles travel along timelike geodesics.
3) Light rays travel along null geodesics (ds = 0).
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Metrics for some 2 and 3 dimensional spaces
To begin let’s review the metrics for some simple spaces. Remember these are the formulas to measure distances in
these spaces for closely spaced points. (We use d instead of ∆ to indicate infinitesimal displacements).
Flat Euclidean 3-space (Cartesian coordinates)
1D ∆d2 = ∆x2
2D ∆d2 = ∆x2 + ∆y 2
3D ∆d2 = ∆x2 + ∆y 2 + ∆z 2
Circle of radius R (S 1 ): ∆d2 = R2 ∆θ2
2D sphere (surface) (S 2 ), (l is length, same as d) Curvature K =
1
R2 :
dl2 = R2 dθ2 + R2 sin2 θdφ2
2D surface of cylinder:
dl2 = dz 2 + R2 dφ2 .
The metric for Euclidean 3-dimensional space (R3 ) in spherical coordinates is
dl2 = dr2 + r2 dθ2 + r2 sin2 θdφ2
Finally the full 4-dimensional flat spacetime metric in spherical coordinates is,
ds2
=
c2 dt2 − dr2 − r2 dθ2 − r2 sin2 θdφ2
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