Metrics in some spherical coordinate systems

Metrics in some spherical coordinate systems Rastko Vukovic, prof. Gimnazija Banja Luka July, 2014 Abstract We consider a pair of spherical coordinate systems. Here are the coordinate transformations relative to the Cartesian system, their Jacobians and linear elements. We mentioned a couple of metrics of the theory of relativity, including Schwartzshild and the Robertson-Walker metric, which is approximately metrics of centrally symmetric gravitational field and our universe. 1 Introduction The point P (ρ, θ, ϕ) in a Spherical coordinate system for three-dimensional space is specified by three numbers: the radial distance ρ (rho) of the point form a fixed origin (point O′ ), its polar angle θ (theta), and azimuthal angle ϕ (phi), as in Figure 1a. Figure 1: a. Spherical (ρ, θ, ϕ), and b. Cylindrical coordinate system On Figure 1b, we see a Cylindrical coordinate system. It has polar axis O′ X and longitudinal axis O′ Z. The point P (ρ, ϕ, z) has radial distance rho, the same angular coordinate phi, and height z. The domains of the coordinates are ρ ∈ [0, ∞), θ ∈ [0, π], ϕ ∈ [0, 2π), and z ∈ (−∞, ∞). 1 R. Vuković Similar to Spherical coordinate system, the Geographic coordinate system uses reverse labels for angles theta and phi. Than the azimuth and elevation of the spherical coordinates locates the point on the Earth, calling them respectively longitude and latitude. The line at which longitude is defined to be 0 is called prime meridian. The line at which latitude is defined by 0 is equator. Plane containing the origin intersects the sphere of constant radius ρ = R in a great circle. These circles are geodesic lines, the shortest paths on the surface of the sphere between two points on a sphere. Each line with constant longitude (meridians) is geodesic line, but the line with constant altitude is not, except the equator. An Alternative spherical coordinate system in which longitudes and altitudes are geodesic lines is shown in Figure 2 on the right. Meridians define the angle ϕ and the central angle subtended at the origin by the given point and equator point of the same meridian is ϑ. In other words, the plane containing the prime meridian rotates about axes North-South for angle phi, and plane containing equator line rotates about axes East-West for angle theta. Figure 2: Coordinates Oxy on a sphere In the figures 2, we can see that the line with the given angle theta is not a geodetic line in a Spherical (left), but is in the Alternative system. However, the lines of constant angle phi (meridian) are geodesic lines in both systems. Coordinate lines on a sphere (ρ = const.) we get by fixing one angle and changing the other. They are mutually perpendicular in the Spherical, but not in the Alternative system. In other words, the Spherical coordinate system is orthogonal, the Alternative is not. 2 R. Vuković 2 Coordinate transformation It is easy to find the transformation of Cartesian rectangular coordinates OXY Z to the Spherical and vice versa: √    ρ = X 2 + Y 2 + Z 2,  X = ρ sin θ cos ϕ, Y = ρ sin θ sin ϕ, (1) θ = arccos (Z/ρ) ,   Z = ρ cos θ; ϕ = arctan (Y /X) . Namely, on Figure 1a, the orthogonal projection of ρ = O′ P to the plane O′ XY is ρz = ρ sin θ, and projection of these to the coordinate axes are X = ρz cos ϕ, Y = ρz sin ϕ. Immediately, the projection of ρ on the third axis is Z = ρ cos θ. Also, for the Cylindrical coordinate system: √    ρ = X 2 + Y 2,  X = ρ cos ϕ, Y = ρ sin ϕ, (2) ϕ = arctan (Y /X) ,   Z = Z; Z = Z. Let’s look at an alternative system in the Figure 3. Denote the distance from the origin by ρ again. Also, the angle ϕ is the same, but the alternate and the spherical angle theta are in relation ϑ = π2 − θ. Figure 3: Alternate spherical coordinate system r, ϕ, ϑ. It is easy to find the coordinate transformation of Descartes and the alternative system: 3 R. Vuković  X = γρ,    Y = γρ tan ϕ, tan ϑ,  Z = γρp   γ = 1/ 1 + tan2 ϕ + tan2 ϑ, √   ρ = X 2 + Y 2 + Z 2, ϕ = arctan(Y /X),  ϑ = arctan(Z/X). (3) Namely, on Figure 3, we see the point P and its projections A, B and C at the coordinate planes X = 0, Y = 0 and Z = 0. We find easily: p p OP = ρ, OC = ρz = X 2 + Y 2 , OB = ρy = X 2 + Z 2 , or X 2 + Y 2 + Z 2 = ρ2 , Y = X tan ϕ, Z = X tan ϑ. Hence the transformation (3). 3 Jacobian matrix From direct transformation (1) left, Spherical to Cartesian, we found differentials:  ∂X ∂X ∂X   dX = ∂ρ dρ + ∂θ dθ + ∂ϕ dϕ, ∂Y ∂Y ∂Y dY = ∂ρ dρ + ∂θ dθ + ∂ϕ dϕ,   dZ = ∂Z dρ + ∂Z dθ + ∂Z dϕ. ∂ρ ∂θ ∂ϕ Then, we find partial derivatives:   dX = sin θ cos ϕdρ + ρ cos θ cos ϕdθ − ρ sin θ sin ϕdϕ, dY = sin θ sin ϕdρ + ρ cos θ sin ϕdθ + ρ sin θ cos ϕdϕ,  dZ = cos θdρ − ρ sin θdθ. (4) (5) Working with these, knowledge of transformation Differential of curvilinear coordinates (5) is almost equally as important as knowing coordinate transformation themselves (1). Generally, equations like (5) we can write in matrix form:     dρ dX dY  = Jˆ  dθ  , (6) dϕ dZ where Jˆ is the Jacobian matrix :  ∂x ∂ρ ∂y ˆ θ, ϕ) =  J(ρ,  ∂ρ ∂z ∂ρ ∂x ∂θ ∂y ∂θ ∂z ∂θ ∂x ∂ϕ ∂y  ∂ϕ  . ∂z ∂ϕ  (7) By calculating the partial derivative of the transformation of spherical coordinates we obtain Jacobian matrix of this (5) transformation:   sin θ cos ϕ ρ cos θ cos ϕ −ρ sin θ sin ϕ (8) JˆS =  sin θ sin ϕ ρ cos θ sin ϕ ρ sin θ cos ϕ  . cos θ −ρ sin θ 0 4 R. Vuković It is easy to calculate the determinant of the matrix: det(JˆS ) = ρ2 sin θ. It is the determinant of Jacobian for spherical system (1). Similarly, Jacobian matrix for the Cylindrical coordinates (2) is:   cos ϕ −ρ sin ϕ 0 JˆC =  sin ϕ ρ cos ϕ 0 . 0 0 1 The corresponding determinant is equal to ρ. Similarly (5), calculate the partial derivatives of alternative system (3): ∂γ γ 3 sin ϕ =− , ∂ϕ cos3 ϕ ∂γ = 0, ∂ρ It is obvious: ∂Y γ sin ϕ = , ∂ρ cos ϕ ∂γ γ 3 sin ϑ =− . ∂ϑ cos3 ϑ ∂Z γ sin ϑ = . ∂r cos ϑ Then it is easy to find: ∂X = γ, ∂ρ ∂X γ 3 ρ sin ϕ =− , ∂ϕ cos3 ϕ ∂X γ 3 ρ sin ϑ =− . ∂ϑ cos3 ϑ ∂Y ∂ ∂γ = (γρ tan ϕ) = (ρ tan ϕ) = ∂ϑ ∂ϑ ∂ϑ =− γ 3 ρ sin ϕ sin ϑ γ 3 sin ϑ · ρ tan ϕ = − . cos3 ϑ cos ϕ cos3 ϑ Similarly, we find symmetrical: γ 3 ρ sin ϕ sin ϑ ∂Z =− . ∂ϕ cos3 ϕ cos ϑ Next, calculate by the product rule: ∂Y ∂ ∂γ ∂ tan ϕ = (γρ tan ϕ) = ρ tan ϕ + γρ = ∂ϕ ∂ϕ ∂ϕ ∂ϕ =− 1 γρ γ 3 sin ϕ · ρ tan ϕ + γρ · = (1 − γ 2 tan2 ϕ) 3 2 cos ϕ cos ϕ cos2 ϕ = γ3ρ γρ 1 + tan2 ϑ = · . 2 2 cos2 ϕ 1 + tan ϕ + tan ϑ cos2 ϕ cos2 ϑ Symmetrical: γ3ρ ∂Z = . ∂ϑ cos2 ϕ cos2 ϑ 5 (9) (10) R. Vuković This is not an orthogonal system of coordinates and the Jacobian is not a diagonal matrix:   2 ρ tan ϑ 2 ρ tan ϕ − γ cos 1 − γ cos 2ϕ 2ϑ   γ2ρ γ 2 ρ tan ϕ tan ϑ  JˆA (ρ, ϕ, ϑ) = γ  (11) tan ϕ − 2 2 2   cos ϕ cos ϑ cos ϑ γ 2 ρ tan ϕ tan ϑ γ2ρ tan ϑ − cos2 ϕ cos2 ϕ cos2 ϑ The determinant of the matrix is: det JˆA = γ 7 ρ2 (1 + tan2 ϕ sin2 ϑ + sin2 ϕ tan2 ϑ − sin2 ϕ sin2 ϑ). cos4 ϕ cos2 ϑ Note, that γ → 1, when ϕ, ϑ → 0, so:   1 0 0 JˆA → 0 ρ 0 , det JˆA → ρ2 , 0 0 ρ ϕ, ϑ → 0. (12) (13) On the surface of a sphere on Figure 2, in the infinitesimal vicinity of the origin O, or near O but for enormous ρ, the coordinate lines are orthogonal. 4 Line element Pythagorean Theorem in the rectangular Cartesian coordinate system reads: dL2 = dX 2 + dY 2 + dZ 2 . (14) Using the differential transformation (5), we have: dL2 = (sin θ cos ϕdρ + ρ cos θ cos ϕdθ − ρ sin θ sin ϕdϕ)2 + +(sin θ sin ϕdρ + ρ cos θ sin ϕdθ + ρ sin θ cos ϕdϕ)2 +(cos θdρ − ρ sin θdθ). After rearranging, we get the Pythagorean Theorem in the Spherical coordinate system: dL2 = dρ2 + ρ2 dθ2 + ρ2 sin2 θdϕ2 , (15) or dL2 = dρ2 + ρ2 dΩ2 , where dΩ2 = dθ2 + sin2 θdϕ2 is the angular measure. In the similar way we can get the Pythagorean theorem in the Cylinder system: dL2 = dρ2 + ρ2 dϕ2 + dZ 2 , (16) or in the Alternate spherical system of coordinates: dL2 = dρ2 + γ 6 ρ2 (1 + tan2 ϕ cos2 ϑ)dϕ2 + (1 + cos2 ϕ tan2 ϑ)dϑ2 cos4 ϕ cos4 ϑ 6 (17) R. Vuković tan ϕ tan ϑ (1 − sin2 ϕ sin2 ϑ)dϕdϑ. cos4 ϕ cos4 ϑ The line element dL has the same length, written in different coordinate systems. On the surface of a sphere, a constant radius ρ = R, line element is −2γ 6 ρ2 dl2 = R2 dΩ2 , (18) for the Spherical coordinates, or similar (dρ = 0) for the Alternate system. Again, when ϕ, ϑ → 0 (or ρ → ∞, near O) the previous sentence (17) can be written approximately: dL2 = dρ2 + ρ2 dϕ2 + ρ2 dϑ2 (ϕ, ϑ → 0). (19) On the surface of a sphere, the constant radius ρ = R, this expression becomes: dl2 = dx2 + dy 2 , (20) where dx = Rdϕ, dy = Rdϑ. 5 Spacetime The idea of space-time has been linked with Albert Einstein’s 1905 theory of special relativity, and the response to the theory by mathematician Hermann Minkowski three years later. The same concept, the union of space and time, Einstein applied (1916) to his general theory of relativity. Here, we observed only two cases of the concept, Schwartzshild metric on the surface of the sphere of a given radius and the exact solution of Einstein’s general equations for homogeneous, isotropic expanding universe. Homogeneous means spatial translation symmetry, isotropic means spatial rotation symmetry. For simplicity, we continue to observe a two-dimensional surface of a sphere of radius ρ = R at time t. In the first case, we have a gravitational field produced by a spherical mass M with its center at the center O′ of our sphere, but we look at a point near the O on the surface of the sphere. As we know, the Schwarzschild metric in spacetime has the form:     Rs 2 2 Rs −1 2 2 ds = 1 − c dt − 1 − dρ − ρ2 (dθ2 + sin2 θdϕ2 ), (21) ρ ρ where c is speed of light, t is the time coordinate (measured by a stationary clock is the Schwarzschild radius of located infinitely far from the massive body), Rs = 2GM c2 the massive body, a scale factor which is related to its mass M , and G = 6.67384 × 10−11 m3 kg−1 s−2 is the gravitational constant. In our case, ρ = R = const., we have   Rs 2 2 2 ds = 1 − c dt − R2 (dθ2 + sin2 θdϕ2 ), R that is 2 ds =  Rs 1− R 7  c2 dt2 − dl2 , (22) R. Vuković where dl2 = R2 (dθ2 + sin2 θdϕ2 ) (23) is the line element on the surface of the sphere. In a special case, in the absence of mass, M → 0 , Eq (21) becomes ds2 = c2 dt2 − dρ2 − ρ2 (dθ2 + sin2 θdϕ2 ). (24) This is infinitesimal interval ds for ordinary Minkowski flat spacetime. We can write ds2 = c2 dt2 − dl2 , wherein at the surface of the sphere is valid (23). If the radius ρ of the sphere increases by constant speed V , then dρ = V dt, so (24) becomes   V2 2 2 2 ds = 1 − 2 c dt − dl2 . (25) c Note that the term (only) formally resembles (22), including the following. Figure 4: Curvatures of our universe. In the second the above-mentioned case, there are different possibilities of spacetime geometries for universe. Friedmann, Robertson and Walker find metric for a flat universe: ds2 = c2 dt2 − a2 (t)[dρ2 + ρ2 (dθ2 + sin2 θdϕ2 )]. A more general version of this metric is given by:   dρ2 2 2 2 2 2 2 2 2 + ρ (dθ + sin θdϕ ) . ds = c dt − a (t) 1 − Kρ2 (26) (27) In the general FRW metric, closed, open and flat universes can be represented by K = +1, −1, 0, as can be seen in Figure 4. According to FRW, the evolution of our universe could be described with respect to the expansion of the universe in time, given by a(t). 8 R. Vuković REFERENCES References [1] Spherical coordinate system, Wikipedia 2014 [2] Cylindrical Coordinates, Wolfram 2014 [3] A. Kumar, General Relativity and Solutions to Einstein’s Field Equations, Department of Physics and Astronomy, Bates College, Lewiston, 2009 [4] A. Mitra, Deriving Friedmann Robertson Walker metric and Hubble’s law from gravitational collapse formalism, Volume 2, 2012, Pages 45–49 9