PHYS 4390: GENERAL RELATIVITY LECTURE 4: FROM SPECIAL TO GENERAL RELATIVITY: GEOMETRY

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1 PHYS 439: GENERA REATIVITY ECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY 1. The path to GR To start our discussion of GR, we must make an assumption about particles interacting with gravity alone. Namely, that in the absence of any other force than gravity, particles will follow a unique worldline in spacetime. Stated differently, if we know the paths of freely-falling objects then we will know gravity. As an example, consider a ball thrown in the air, according to GR the ball is following a straigth path in the curved spacetime produced by the Earth. Figure 1. Parabola of a ball thrown into the air. For much of the history of geometry, the Euclidean space was studied. These arise from Euclid s postulates and from these we may derive the properties of the D plane. It was only in 183 when Janos Bolyai and Nicolai obachevsky independently proved that the fifth postulate does not follow from the first four and allowed for the consideration of non-euclidean geometries 1 (Of course, Gauss alrea knew of this as well, but he was not a fan of these geometries.) If instead we consider geometry as an experimental science, what would we measure? Circumference of a circle compared to its radius. The sum of angles in a triangle. The (possible) intersection of lines that are initially parallel. Is the geometry the same in all directions? (Isotropy) Is the geometry the same in all places? (Homogeneity) 1 Spheres, hyperbolic planes, more general D surfaces in 3D Euclidean space, the sunken city of R lyeh, etc... 1

2 PHYS 439: GENERA REATIVITY ECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY For a D plane we have: C πr. The sum of angles in a triangle is π. There are no intersecting parallel lines. space is homogeneous and isotropic. Of course we know all of this holds because we can check Euclid s five postulates.. Riemannian Geometry: The D Plane R The best way to stu the geometry of more general surfaces, is by measuring how distances between nearby points change. This is the starting point of Riemannian Geometry which is a subset of differential geometry. In the case of Euclidean geometry, the line element is ds or equivalently ds R distance between nearby points (dx + ) 1. This was relative to the Cartesian coordinate system, we may rewrite this in a different coordinate system. For example, the polar coordinates x r cos φ, y r sin φ yield ds R (dr + r dφ ) 1 Figure. Polar Coordinates To determine homogeniety, translate the Cartesian coordinates and consider the differentials of x and y, x x + a, ỹ y + b d x dx, dỹ, thus d s R ds R. For isotropy we rotate the Cartesian coordinates, [ ] [ ] [ ] x cos θ sin θ x, ỹ sin θ cos θ y by computing the differentials and using a simple trigonometric identity, we may show that d s R ds R.

3 PHYS 439: GENERA REATIVITYECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY3 To determine the circumference of a circle, suppose x + y R for some fixed value R, then C ds (dx + ) 1 R ( dx 1 + R R R ( )) 1 dx R dx R x, x +y R since x + y R implying that xdx + y. Changing variables x Rζ this is 1 dζ C R πr 1 1 ζ of course this is much easier to prove in polar coordinates, since C ds π Rdφ πr. 3. Riemannian Geometry: The -sphere S As a simple example of a non-euclidean geometry, we will consider the surface of a sphere of radius a with polar coordinates (θ, φ) to label points on the sphere. What will the line element look like? Figure 3. Measuring points on a sphere Combining the two differentials in the diagram ds S a (dθ + sin θdφ ). We will define circles as the loci of points on the sphere which are a fixed distance from another point. We may orient our coordinate system so that the polar axis is centered on the circle. For a circle θ Θ Constant the differential dθ and so π C ds a sin θdφ πa sin θ

4 4 PHYS 439: GENERA REATIVITY ECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY Figure 4 The radius is the distance from the point to the circle Z circle Z Θ r ds adθ aθ center Solving for θ we find C r πa sin a 1 r πr a For large ar, the inverse will be small and this reduces to Euclidean geometry. The following facts may be proven in a similar manner Sum of angles in a triangle is π + area a Parallel lines meet in distance πa ine element is homogeneous and isotropy. The circumference of circles reaches a maximum of θ Figure 5 π

5 PHYS 439: GENERA REATIVITYECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY5 4. Coordinate transformations on the Sphere Under a general coordiante transformation the differentials become θ f( θ, φ), φ g( θ, φ) dθ f θ d θ + f d φ φ dφ g θ d θ + g d φ φ Substituting these into the line element produces [( ( f ) ( ) ) ds a θ + sin (f( θ, φ)) g φ d θ +...]. This illustrates that even in two dimensions a general coordinate transformation is a non-trivial affair for the line element. While we cannot map ds S ds R we may use a coordinate transformation to locally make the geometry of the sphere resemble that of a plane. The analogy to map making is unavoidable; in fact map making could be seen as a special case of differential geometry. In GR, we will want to consider this for more general surfaces, or in the case of Penrose diagrams, map the infinite structure of spacetime into something finite. As an example, we will map the sphere into a plane, using longitude φ and lattitude λ π θ so that sin θ cos λ and the line element becomes ds a (dθ + sin θdφ ) a dλ + cos λdφ ) Now introduce the new coordinates x x(λ, φ), y y(λ, φ), then different functions for x and y produce different projections. Example 4.1. Consider the mapping, x φ π, y λ, where is the size of map; π the range of x and y will be and respectively. Substituting for dλ and dφ the line element becomes ( πa ) ( ( ds + cos πy ) dx ) This transformation will preserve the angles on a sphere when mapped into the plane.

6 6 PHYS 439: GENERA REATIVITY ECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY Figure 6 To see this, consider the displacement in the y-direction and x-direction respectively πy πa πa dsx cos dx. dsy Taking the ratio, we find that dsy tan Ψ dsx cos πy dx If y lies along the equator, then y and cos πy 1. Thus the angles are preserved along the equator, outside of this region, the angles are not preserved. We will now consider a projection for which angles are preserved. Example 4.. Consider the mapping φ x, y y(λ) π with y chosen so that λ λ(y) exists. The differentials are then π dλ dx, and dλ upon substitution into the line element this is " # π dλ (1) ds a cos(λ(y)) dx + dφ In 1569 Gerard de Kremer, or as he is more commonly known, Gerard Mercator proved that it is possible to choose λ(y) so that the line element preserves angles the example he found is known as the Mercator projection. To preserve angles, we must have tan Ψ dx, and so any line element of the form ds Ω (x, y)[dx + ] will work just fine. Returning to (1) if we assume π dλ cos(λ(y)) we may invert this and solve the resulting ordinary differential equation for y(λ) Z λ dλ π λ y(λ) ln tan +. cos λ π 4

7 PHYS 439: GENERA REATIVITYECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY7 Together with x φ π this defines the Mercator projection, and the effect of the transformation on the line element is [ ] πa ds cos(λ(y)) (dx + ). While angles are preserved, areas will change since ds x ds y (Ω(y) x)(ω(y) y) Ω (y) x y This explains why Greenland looks so large when compared with South America - as one travels towards the poles the Ω(y) factor goes to zero, and the y position becomes more extended. Example 4.3. As one more example of a more general surface than the sphere, consider the peanut geometry outlined in Hartle, ds a (dθ + f (θ)dφ ) where f(θ) sin θ ( sin θ ). The circumference is then π ( C af(θ)dφ πaf(θ) πa sin θ 1 3 ) 4 sin θ while the pole to pole distance is a π dθ πa Figure 7 5. The -Sphere as a Manifold With our discussion of the sphere complete, we can use our exercise in mapping the sphere into the plane to tentatively introduce the idea of a manifold. We have seen some mappings that take regions of the sphere and map them into a subset of the plane. This approach of transforming regions of the non-euclidean geometry into something that looks Euclidean will allow us to extend the vector calculus to more general geometries. Without getting into the finer points of manifolds, like topological spaces and open sets, we can imagine a manifold as a set of subsets of the original manifold that can be mapped to R n (n is the dimension of the manifold), and that these subsets taken together describe the original manifold in a smooth way. In more formal terms, an n-dimensional C real manifold M is a set of mappings together with a collection of subsets {O α } satisfying the following properties Each p M lies in at least one O α

8 8PHYS 439: GENERA REATIVITY ECTURE 4: FROM SPECIA TO GENERA REATIVITY: GEOMETRY For each α there is a one-to-one map Ψ α : O α U α where U a is an open subset of R n. If any two sets O α and O β overlap, O α O β, and the map Ψ β Ψ 1 α taking points in Ψ α (O α O β ) U α R n to points Ψ β (O α O β ) U β R n with the requirement that the composition of these functions are also infinitely continuously differentiable. Example 5.1. As a very simple example of a manifold, consider S with stereographic projection. To cover the whole manifold one must use two coordinate charts: from the north and south pole respectively. Thus we have {O α } {S \ {N}, S \ {S}} where N and S denote the point at the north and south pole of the sphere. With the chain rule, one may show that any composition of the pair of stereographic projections is C, i.e., infinitely continuously differentiable. Mathematicians call the each ma,p Ψ α, a chart (or a coordinate system), and the entire set of charts, {Ψ α }, an atlas. For our interests, coordinate system is more appropriate. To define a manifold in a unique way, often it is required in the definition of a manifold, that the cover {O α } and the atlas {Ψ α } is maximal - that is, all coordinate systems compatible with properties and 3 are included.