Part V – Coordinates and Formal Analysis

5 Coordinate Systems

5.1 Rectangular (Cartesian) Coordinate System

To distinguish between points in space, a coordinate system is introduced. The most important characteristics of a coordinate system are the origin and the coordinate axes. The origin may be chosen for convenience, and the axes are usually taken to be Cartesian because of their mathematical simplicity.

In a Cartesian coordinate system:

• The axes are perpendicular (orthogonal) to each other.
• The axes are independent; changing one coordinate does not affect the others.
• The axes have direction and magnitude and can therefore be treated as vectors.

A point in space is represented by its coordinates, for example \( A(x_a, y_a) \). The value \( x_a \) is found by drawing a line parallel to the y‑axis; where it intersects the x‑axis is the coordinate \( x_a \). The same applies to \( y_a \).

The distance from point A to the origin follows from Pythagoras: \[ |A - O_{\text{origin}}|^{2} = x_a^{2} + y_a^{2}. \]

For a line segment between A and B, the length is: \[ |A - B|^{2} = (x_a - x_b)^{2} + (y_a - y_b)^{2}. \]

The advantage is that the length of the line segment is independent of the arbitrarily chosen origin; the coordinate values change, but the difference \( |A - B| \) does not.

5.2 Non‑Orthogonal Coordinate System

For practical reasons, one may also choose a coordinate system whose axes are not orthogonal. Such a system can still describe positions and distances, but the calculations become slightly more involved.

A line segment \( s \) in such a system is the sum of the basis vectors: \[ \vec{s} = x\,\vec{a} + y\,\vec{b}. \]

The magnitude \( s \) of \( \vec{s} \) follows from the inner product: \[ \vec{s}\cdot\vec{s} = (x\vec{a} + y\vec{b}) \cdot (x\vec{a} + y\vec{b}) = x^{2}(\vec{a}\cdot\vec{a}) + xy(\vec{a}\cdot\vec{b}) + xy(\vec{b}\cdot\vec{a}) + y^{2}(\vec{b}\cdot\vec{b}). \]

Since \( \vec{a}\cdot\vec{a} = 1 \), \( \vec{b}\cdot\vec{b} = 1 \), and \( \vec{a}\cdot\vec{b} = \cos\alpha \), we obtain: \[ s^{2} = x^{2} + 2xy\cos\alpha + y^{2}. \]

If \( \varphi \) is the angle between \( x \) and \( y \), then: \[ \cos\alpha = -\cos\varphi, \qquad \cos\alpha = \cos(180^\circ - \varphi) = -\cos\varphi. \]

Thus: \[ s^{2} = x^{2} + y^{2} - 2xy\cos\varphi. \]

This is the familiar cosine rule. Besides the squares of the coordinates, the product of the coordinates also appears.

5.3 Curved Coordinates

Instead of non‑orthogonal axes, it may be useful to employ curved coordinates. Working with such coordinates is more complex, but Einstein used the following idea:

A curved line can be regarded as composed of infinitely many small straight segments. By examining an infinitesimal region, these curved coordinates behave like a local linear coordinate system, though not necessarily orthogonal.

Because the coordinate system is infinitesimal, the coordinates are written as \( dx, dy \), etc. These coordinates have coefficients that encode the curvature of the coordinate system. In curved space, these coefficients are no longer constant but depend on position.

Gravity is said to bend coordinate systems and thereby deform spacetime, creating a gravitational field and causing acceleration. By choosing a curved coordinate system that moves and bends with the gravitational field, no force is experienced—analogous to choosing a moving coordinate system in special relativity to neutralize velocity.

5.4 General Form of a Coordinate System

Let us derive a relation between a line segment and its curved coordinate system.

As noted earlier, an infinitesimal line segment \( d\vec{s} \) is a vector, and its magnitude is: \[ d\vec{s}\cdot d\vec{s} = (dx + dy)\cdot(dx + dy) = dx\cdot dx + dx\cdot dy + dy\cdot dx + dy\cdot dy, \] for a linear, non‑orthogonal system.

To obtain a more general form (not necessarily orthogonal), each term is given a coefficient \( g_{\mu\nu} \): \[ ds^{2} = g_{xx}\,dx\,dx + g_{xy}\,dx\,dy + g_{yx}\,dy\,dx + g_{yy}\,dy\,dy. \]

In the cosine‑rule example above: \[ g_{xx} = g_{yy} = 1, \qquad g_{xy} = g_{yx} = -\cos\varphi. \]

The coefficients \( g_{\mu\nu} \) form the metric tensor, which in this two‑dimensional case is: \[ g_{\mu\nu} = \begin{pmatrix} 1 & -\cos\varphi \\ -\cos\varphi & 1 \end{pmatrix}. \]

5.5 The Metric Tensor and Einstein Notation

For a general four‑dimensional spacetime (with time coordinate \( ct \)), the metric is a 4×4 tensor. The general form is: \[ ds^{2} = \sum_{\mu=0}^{3}\sum_{\nu=0}^{3} g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}. \]

In Einstein notation: \[ ds^{2} = g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}. \]

Here \( \mu \) and \( \nu \) run from 0 to 3, with: \[ x^{0} = ct,\qquad x^{1} = x,\qquad x^{2} = y,\qquad x^{3} = z. \] The metric tensor \( g_{\mu\nu} \) contains all information about the curvature of spacetime.

Example of a metric tensor in matrix form:

\[ g_{\mu\nu} = \begin{pmatrix} g_{00} & g_{01} & g_{02} & g_{03} \\ g_{10} & g_{11} & g_{12} & g_{13} \\ g_{20} & g_{21} & g_{22} & g_{23} \\ g_{30} & g_{31} & g_{32} & g_{33} \end{pmatrix}. \]

If the coordinate system is orthogonal, all cross‑terms vanish: \[ g_{\mu\nu} = 0 \quad \text{for } \mu \neq \nu. \]

The value of \( ds^{2} \) remains unchanged under a coordinate transformation, provided the metric transforms accordingly: \[ ds^{2} = g_{\mu\nu}(x)\,dx^{\mu}\,dx^{\nu} = g_{\alpha\beta}(y)\,dy^{\alpha}\,dy^{\beta}. \]

Thus, \( g_{\mu\nu} \) acts as the “weight factor” determining how infinitesimal displacements contribute to distance.

• The diagonal elements \( g_{\mu\mu} \) act as scale factors for each coordinate direction.
• The off‑diagonal elements \( g_{\mu\nu} \) with \( \mu \neq \nu \) describe whether coordinate directions are skew (non‑orthogonal), analogous to direction cosines.

Summary

• A coordinate system structures space and allows distances to be computed.
• In orthogonal systems, Pythagoras applies; in non‑orthogonal systems, the cosine rule applies.
• Curved coordinate systems are essential for describing gravitational fields in general relativity.
• The metric \( g_{\mu\nu} \) contains all information about distance measurement and curvature of space or spacetime.

5.6 Transformation Between Two Coordinate Systems

As mentioned earlier, in a curved coordinate system one may use, “locally,” within an infinitesimally small region, a coordinate system with straight lines. For a four‑dimensional coordinate system, each new coordinate in the new \( x \)-system has a linear relation with all old coordinates in the old \( y \)-system, according to: \[ dx^{0} = \frac{\partial x^{0}}{\partial y^{0}}\,dy^{0} + \frac{\partial x^{0}}{\partial y^{1}}\,dy^{1} + \frac{\partial x^{0}}{\partial y^{2}}\,dy^{2} + \frac{\partial x^{0}}{\partial y^{3}}\,dy^{3}. \]

The same applies to the three other coordinates, which leads to the general formula: \[ dx^{m} = \frac{\partial x^{m}}{\partial y^{r}}\,dy^{r}. \]

Summation takes place over the repeated index \( r \). Here, summation over the index \( r \) (from 0 to 3) is implied by Einstein notation. This means that for each value of \( m \), the derivatives over all values of \( r \) are added. This formula describes how an infinitesimal change in the new coordinate system \( x^{m} \) is built from changes in the old system \( y^{r} \).

5.6.1 Extended Explanation of the Metric Tensor

We begin with a Cartesian coordinate system, which in this case is comparable to the Minkowski expression (see Section 5.10.1 and Appendix 9.1, equation 11a) in special relativity: \[ ds^{2} = c^{2}dt^{2} - dx^{2} - dy^{2} - dz^{2}. \]

To make the notation more compact and general, we rename the differential terms: \[ cdt = dx^{0},\qquad dx = dx^{1},\qquad dy = dx^{2},\qquad dz = dx^{3}. \] All differential terms now have the dimension of length (meters).

In this notation, the space‑time interval becomes: \[ ds^{2} = (dx^{0})^{2} - (dx^{1})^{2} - (dx^{2})^{2} - (dx^{3})^{2} = \eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}. \]

The metric tensor \( \eta_{\mu\nu} \) (the Minkowski metric) is, in matrix form: \[ \eta_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & -1 & -1 \end{pmatrix}. \]

This tensor describes the distance structure of flat space‑time. The distance between two events is therefore: \[ ds^{2} = \eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}. \]

Now we consider an arbitrary coordinate system \( y^{\alpha} \), with coordinates \( y^{0}, y^{1}, y^{2}, y^{3} \). The relation between the old and new systems is given by the chain rule: \[ dx^{\mu} = \frac{\partial x^{\mu}}{\partial y^{0}}\,dy^{0} + \frac{\partial x^{\mu}}{\partial y^{1}}\,dy^{1} + \frac{\partial x^{\mu}}{\partial y^{2}}\,dy^{2} + \frac{\partial x^{\mu}}{\partial y^{3}}\,dy^{3}. \]

Or in compact notation: \[ dx^{\mu} = \frac{\partial x^{\mu}}{\partial y^{\alpha}}\,dy^{\alpha}, \qquad dx^{\nu} = \frac{\partial x^{\nu}}{\partial y^{\beta}}\,dy^{\beta}. \]

Substituting into the Minkowski form: \[ ds^{2} = \eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu} \] yields: \[ ds^{2} = \eta_{\mu\nu} \frac{\partial x^{\mu}}{\partial y^{\alpha}} \frac{\partial x^{\nu}}{\partial y^{\beta}} \,dy^{\alpha}\,dy^{\beta}. \]

We now define a new metric tensor \( g_{\alpha\beta} \) in the coordinate system \( y^{\alpha} \) as follows: \[ g_{\alpha\beta} = \eta_{\mu\nu} \frac{\partial x^{\mu}}{\partial y^{\alpha}} \frac{\partial x^{\nu}}{\partial y^{\beta}}. \]

So that: \[ ds^{2} = g_{\alpha\beta}\,dy^{\alpha}\,dy^{\beta}. \]

If we then move to another arbitrary coordinate system \( x^{\mu} \), the inverse transformation holds: \[ ds^{2} = g_{\alpha\beta} \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\nu}} \,dx^{\mu}\,dx^{\nu} = g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}. \]

From this follows the general transformation formula for the metric tensor: \[ g_{\mu\nu}(x) = \frac{\partial y^{\alpha}}{\partial x^{\mu}} \frac{\partial y^{\beta}}{\partial x^{\nu}} g_{\alpha\beta}(y). \] This formula describes how the components of the metric tensor transform under a general coordinate transformation. It is a fundamental result in general relativity and forms the basis for understanding curved space‑time.

5.7 Transformation Between Cartesian and Polar (Infinitesimal) Coordinates

As an example, we now perform the transformation from a Cartesian to a spherical (polar) coordinate system. We assume the reader is familiar with the standard transformation between the two systems: \[ x = r\sin\theta\cos\varphi,\qquad y = r\sin\theta\sin\varphi,\qquad z = r\cos\theta. \]

Derivation of \( dx, dy, dz \)

We differentiate the above expressions to obtain the infinitesimal displacements: \[ d\vec x = \sin\theta\cos\varphi\,d\vec r + r\cos\theta\cos\varphi\,d\vec \theta - r\sin\theta\sin\varphi\,d\vec \varphi \]\[ d\vec y = \sin\theta\sin\varphi\,d\vec r + r\cos\theta\sin\varphi\,d\vec \theta + r\sin\theta\cos\varphi\,d\vec \varphi \]\[ d\vec z = \cos\theta\,d\vec r - r\sin\theta\,d\vec \theta. \]

These vector differentials describe the infinitesimal displacements in the \( x \)-, \( y \)- and \( z \)-directions in terms of \( dr, d\theta, d\varphi \).

Determining the Squares of the Differentials

To determine the magnitude of \( dx \), \( dy \), and \( dz \), we take the inner product of each: \[ dx^{2} = d\vec x \cdot d\vec x,\qquad dy^{2} = d\vec y \cdot d\vec y,\qquad dz^{2} = d\vec z \cdot d\vec z. \]

Because the coordinates \( r, \theta, \varphi \) are orthogonal, the cross‑terms vanish, resulting in: \[ dx^{2} = \sin^{2}\theta\cos^{2}\varphi\,dr^{2} + r^{2}\cos^{2}\theta\cos^{2}\varphi\,d\theta^{2} + r^{2}\sin^{2}\theta\sin^{2}\varphi\,d\varphi^{2}, \] \[ dy^{2} = \sin^{2}\theta\sin^{2}\varphi\,dr^{2} + r^{2}\cos^{2}\theta\sin^{2}\varphi\,d\theta^{2} + r^{2}\sin^{2}\theta\cos^{2}\varphi\,d\varphi^{2}, \] \[ dz^{2} = \cos^{2}\theta\,dr^{2} + r^{2}\sin^{2}\theta\,d\theta^{2}. \]

Summing \( dx^{2} + dy^{2} + dz^{2} \)

Adding the three expressions gives: \[ dx^{2} + dy^{2} + dz^{2} = \sin^{2}\theta\cos^{2}\varphi\,dr^{2} + \sin^{2}\theta\sin^{2}\varphi\,dr^{2} + \cos^{2}\theta\,dr^{2} \] \[ \quad + r^{2}\cos^{2}\theta\cos^{2}\varphi\,d\theta^{2} + r^{2}\cos^{2}\theta\sin^{2}\varphi\,d\theta^{2} + r^{2}\sin^{2}\theta\,d\theta^{2} \] \[ \quad + r^{2}\sin^{2}\theta\sin^{2}\varphi\,d\varphi^{2} + r^{2}\sin^{2}\theta\cos^{2}\varphi\,d\varphi^{2}. \]

Using the trigonometric identities: \[ \cos^{2}\varphi + \sin^{2}\varphi = 1, \qquad \cos^{2}\theta + \sin^{2}\theta = 1, \] this simplifies to: \[ dx^{2} + dy^{2} + dz^{2} = dr^{2} + r^{2}d\theta^{2} + r^{2}\sin^{2}\theta\,d\varphi^{2}. \tag{1} \]

This expression is exactly the spatial component of the metric in spherical coordinates. The time component may be added as: \[ ds^{2} = c^{2}dt^{2} - dx^{2} - dy^{2} - dz^{2}, \] or in spherical form: \[ ds^{2} = c^{2}dt^{2} - dr^{2} - r^{2}d\theta^{2} - r^{2}\sin^{2}\theta\,d\varphi^{2}. \]

Volume Element in Spherical Coordinates

This describes the transformation from a Cartesian coordinate system to a spherical (polar) coordinate system.

Volume Element in Cartesian and Spherical Coordinates

The volume element in Cartesian coordinates is: \[ dV = dx\,dy\,dz. \]

After transformation to spherical coordinates, this becomes: \[ dV = dr \cdot (r\,d\theta) \cdot (r\sin\theta\,d\varphi) = r^{2}\sin\theta\,dr\,d\theta\,d\varphi. \]

Computing the Volume of a Sphere

The total volume of a sphere of radius \( R \) follows from the integral: \[ V = \iiint r^{2}\sin\theta\,dr\,d\theta\,d\varphi, \] with integration limits:

• \( r \in [0, R] \)
• \( \theta \in [0, \pi] \)
• \( \varphi \in [0, 2\pi] \)

The integral becomes: \[ V = \left( \int_{0}^{R} r^{2}\,dr \right) \left( \int_{0}^{\pi} \sin\theta\,d\theta \right) \left( \int_{0}^{2\pi} d\varphi \right). \]

Evaluation: \[ \int_{0}^{R} r^{2}\,dr = \frac{1}{3}R^{3}, \qquad \int_{0}^{\pi} \sin\theta\,d\theta = [-\cos\theta]_{0}^{\pi} = 2, \qquad \int_{0}^{2\pi} d\varphi = 2\pi. \]

Thus: \[ V = \frac{1}{3}R^{3} \cdot 2 \cdot 2\pi = \frac{4}{3}\pi R^{3}. \]

This confirms the well‑known result for the volume of a sphere.

5.8 Exercise: Applying the Metric Transformation Formula

Here we show how the metric transformation formula is formally applied when moving from a Cartesian to a polar (spherical) coordinate system.

1. General Formulas

We recall the following relations:

1.1 Transformation of Coordinates

\[ dx^{m} = \frac{\partial x^{m}}{\partial y^{r}}\,dy^{r}. \]

1.2 Line Element in Cartesian Coordinates

\[ ds^{2} = \eta_{mn}\,d\xi^{m}\,d\xi^{n}. \]

1.3 Invariance of the Line Element Under Coordinate Transformation

\[ ds^{2} = g_{mn}(x)\,dx^{m}\,dx^{n} = g_{pq}(y)\,dy^{p}\,dy^{q}. \]

1.4 Transformation Formula for the Metric

\[ g_{pq}(y) = g_{mn}(x)\, \frac{\partial x^{m}}{\partial y^{p}}\, \frac{\partial x^{n}}{\partial y^{q}}. \]

2. From Cartesian to Spherical

We consider the following Cartesian metric (in four‑dimensional spacetime with signature \( (+,-,-,-) \)): \[ ds^{2} = c^{2}dt^{2} - dx^{2} - dy^{2} - dz^{2}. \]

The corresponding form in spherical coordinates is (using equation 4.6.1): \[ ds^{2} = c^{2}dt^{2} - dr^{2} - r^{2}d\theta^{2} - r^{2}\sin^{2}\theta\,d\varphi^{2}. \]

Metric in Cartesian Coordinates

The metric tensor in Cartesian coordinates is: \[ g_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}. \]

Thus: \[ g_{00} = 1,\qquad g_{11} = -1,\qquad g_{22} = -1,\qquad g_{33} = -1, \] and all remaining elements are zero.

Objective

Find the metric \( g_{\mu\nu} \) in spherical coordinates, namely: \[ g_{00} = 1,\qquad g_{11} = -1,\qquad g_{22} = -r^{2},\qquad g_{33} = -r^{2}\sin^{2}\theta. \]

3. Coordinate Transformation

The spherical coordinates are expressed as functions of the Cartesian coordinates: \[ x = r\sin\theta\cos\varphi,\qquad y = r\sin\theta\sin\varphi,\qquad z = r\cos\theta. \]

We apply the transformation formula: \[ dx^{m} = \frac{\partial x^{m}}{\partial y^{r}}\,dy^{r}. \]

When fully expanded, this becomes: \[ dt = \frac{\partial t}{\partial t}\,dt + \frac{\partial t}{\partial r}\,dr + \frac{\partial t}{\partial \theta}\,d\theta + \frac{\partial t}{\partial \varphi}\,d\varphi, \] \[ dx = \frac{\partial x}{\partial t}\,dt + \frac{\partial x}{\partial r}\,dr + \frac{\partial x}{\partial \theta}\,d\theta + \frac{\partial x}{\partial \varphi}\,d\varphi, \] \[ dy = \frac{\partial y}{\partial t}\,dt + \frac{\partial y}{\partial r}\,dr + \frac{\partial y}{\partial \theta}\,d\theta + \frac{\partial y}{\partial \varphi}\,d\varphi, \] \[ dz = \frac{\partial z}{\partial t}\,dt + \frac{\partial z}{\partial r}\,dr + \frac{\partial z}{\partial \theta}\,d\theta + \frac{\partial z}{\partial \varphi}\,d\varphi. \]

The Differentials Become

\[ dt = dt, \] \[ dx = \sin\theta\cos\varphi\,dr + r\cos\theta\cos\varphi\,d\theta - r\sin\theta\sin\varphi\,d\varphi, \] \[ dy = \sin\theta\sin\varphi\,dr + r\cos\theta\sin\varphi\,d\theta + r\sin\theta\cos\varphi\,d\varphi, \] \[ dz = \cos\theta\,dr - r\sin\theta\,d\theta. \]

Thus the Jacobian Elements Are

\[ \frac{\partial t}{\partial t} = 1,\qquad \frac{\partial t}{\partial r} = 0,\qquad \frac{\partial t}{\partial \theta} = 0,\qquad \frac{\partial t}{\partial \varphi} = 0, \] \[ \frac{\partial x}{\partial t} = 0,\qquad \frac{\partial x}{\partial r} = \sin\theta\cos\varphi,\qquad \frac{\partial x}{\partial \theta} = r\cos\theta\cos\varphi,\qquad \frac{\partial x}{\partial \varphi} = -r\sin\theta\sin\varphi, \] \[ \frac{\partial y}{\partial t} = 0,\qquad \frac{\partial y}{\partial r} = \sin\theta\sin\varphi,\qquad \frac{\partial y}{\partial \theta} = r\cos\theta\sin\varphi,\qquad \frac{\partial y}{\partial \varphi} = r\sin\theta\cos\varphi, \] \[ \frac{\partial z}{\partial t} = 0,\qquad \frac{\partial z}{\partial r} = \cos\theta,\qquad \frac{\partial z}{\partial \theta} = -r\sin\theta,\qquad \frac{\partial z}{\partial \varphi} = 0. \]

4. Computing the New Metric

We now apply the transformation formula: \[ g_{pq}(y) = g_{mn}(x)\, \frac{\partial x^{m}}{\partial y^{p}}\, \frac{\partial x^{n}}{\partial y^{q}}. \]

The metric tensor components expand to: \[ g_{00}(y) = g_{00}(x)\frac{\partial x^{0}}{\partial y^{0}}\frac{\partial x^{0}}{\partial y^{0}} + g_{01}(x)\frac{\partial x^{0}}{\partial y^{0}}\frac{\partial x^{1}}{\partial y^{0}} + g_{02}(x)\frac{\partial x^{0}}{\partial y^{0}}\frac{\partial x^{2}}{\partial y^{0}} + g_{03}(x)\frac{\partial x^{0}}{\partial y^{0}}\frac{\partial x^{3}}{\partial y^{0}} \] \[ +\, g_{10}(x)\frac{\partial x^{1}}{\partial y^{0}}\frac{\partial x^{0}}{\partial y^{0}} + g_{11}(x)\frac{\partial x^{1}}{\partial y^{0}}\frac{\partial x^{1}}{\partial y^{0}} + g_{12}(x)\frac{\partial x^{1}}{\partial y^{0}}\frac{\partial x^{2}}{\partial y^{0}} + g_{13}(x)\frac{\partial x^{1}}{\partial y^{0}}\frac{\partial x^{3}}{\partial y^{0}} \] \[ +\, g_{20}(x)\frac{\partial x^{2}}{\partial y^{0}}\frac{\partial x^{0}}{\partial y^{0}} + g_{21}(x)\frac{\partial x^{2}}{\partial y^{0}}\frac{\partial x^{1}}{\partial y^{0}} + g_{22}(x)\frac{\partial x^{2}}{\partial y^{0}}\frac{\partial x^{2}}{\partial y^{0}} + g_{23}(x)\frac{\partial x^{2}}{\partial y^{0}}\frac{\partial x^{3}}{\partial y^{0}} \] \[ +\, g_{30}(x)\frac{\partial x^{3}}{\partial y^{0}}\frac{\partial x^{0}}{\partial y^{0}} + g_{31}(x)\frac{\partial x^{3}}{\partial y^{0}}\frac{\partial x^{1}}{\partial y^{0}} + g_{32}(x)\frac{\partial x^{3}}{\partial y^{0}}\frac{\partial x^{2}}{\partial y^{0}} + g_{33}(x)\frac{\partial x^{3}}{\partial y^{0}}\frac{\partial x^{3}}{\partial y^{0}}. \]

Example: the Radial Component \( g_{rr} \)

We substitute the appropriate spherical and Cartesian derivatives: \[ g_{rr} = g_{tt}\frac{\partial t}{\partial r}\frac{\partial t}{\partial r} + g_{xx}\frac{\partial x}{\partial r}\frac{\partial x}{\partial r} + g_{yy}\frac{\partial y}{\partial r}\frac{\partial y}{\partial r} + g_{zz}\frac{\partial z}{\partial r}\frac{\partial z}{\partial r}. \]

Since the Cartesian system is orthogonal, only diagonal elements are non‑zero: \[ g_{tt}=1,\qquad g_{xx}=-1,\qquad g_{yy}=-1,\qquad g_{zz}=-1. \]

The required derivatives are: \[ \frac{\partial t}{\partial r}=0,\qquad \frac{\partial x}{\partial r}=\sin\theta\cos\varphi,\qquad \frac{\partial y}{\partial r}=\sin\theta\sin\varphi,\qquad \frac{\partial z}{\partial r}=\cos\theta. \]

Thus: \[ g_{rr} = 1\cdot 0^{2} - (\sin\theta\cos\varphi)^{2} - (\sin\theta\sin\varphi)^{2} - (\cos\theta)^{2}. \]

Using: \[ \cos^{2}\varphi + \sin^{2}\varphi = 1,\qquad \sin^{2}\theta + \cos^{2}\theta = 1, \] we obtain: \[ g_{rr} = -\left[\sin^{2}\theta(\cos^{2}\varphi + \sin^{2}\varphi) + \cos^{2}\theta\right] = -(\sin^{2}\theta + \cos^{2}\theta) = -1. \]

Thus: \[ g_{rr} = -1. \]

1.5 Time Component

\[ g_{tt} = g_{tt}\left(\frac{\partial t}{\partial t}\right)^{2} + g_{xx}\left(\frac{\partial x}{\partial t}\right)^{2} + g_{yy}\left(\frac{\partial y}{\partial t}\right)^{2} + g_{zz}\left(\frac{\partial z}{\partial t}\right)^{2}. \]

Since: \[ \frac{\partial t}{\partial t}=1,\qquad \frac{\partial x}{\partial t}=0,\qquad \frac{\partial y}{\partial t}=0,\qquad \frac{\partial z}{\partial t}=0, \] we find: \[ g_{tt} = 1. \]

1.6 Radial Component

\[ g_{rr} = g_{tt}\left(\frac{\partial t}{\partial r}\right)^{2} + g_{xx}\left(\frac{\partial x}{\partial r}\right)^{2} + g_{yy}\left(\frac{\partial y}{\partial r}\right)^{2} + g_{zz}\left(\frac{\partial z}{\partial r}\right)^{2} = -1. \]

Explicitly: \[ g_{rr} = 0 - (\sin\theta\cos\varphi)^{2} - (\sin\theta\sin\varphi)^{2} - (\cos\theta)^{2}. \]

Using: \[ \cos^{2}\varphi + \sin^{2}\varphi = 1,\qquad \sin^{2}\theta + \cos^{2}\theta = 1, \] we obtain: \[ g_{rr} = -\left[\sin^{2}\theta(\cos^{2}\varphi + \sin^{2}\varphi) + \cos^{2}\theta\right] = -1. \]

1.7 Angular Component \( \theta \)

\[ g_{\theta\theta} = g_{tt}\left(\frac{\partial t}{\partial \theta}\right)^{2} + g_{xx}\left(\frac{\partial x}{\partial \theta}\right)^{2} + g_{yy}\left(\frac{\partial y}{\partial \theta}\right)^{2} + g_{zz}\left(\frac{\partial z}{\partial \theta}\right)^{2}. \]

With: \[ \frac{\partial x}{\partial \theta} = r\cos\theta\cos\varphi,\qquad \frac{\partial y}{\partial \theta} = r\cos\theta\sin\varphi,\qquad \frac{\partial z}{\partial \theta} = -r\sin\theta, \] we obtain: \[ g_{\theta\theta} = -\left[ (r\cos\theta\cos\varphi)^{2} + (r\cos\theta\sin\varphi)^{2} + (-r\sin\theta)^{2} \right]. \]

\[ g_{\theta\theta} = -r^{2}\left[\cos^{2}\theta(\cos^{2}\varphi + \sin^{2}\varphi) + \sin^{2}\theta\right] = -r^{2}. \]

1.8 Angular Component \( \varphi \)

\[ g_{\varphi\varphi} = g_{tt}\left(\frac{\partial t}{\partial \varphi}\right)^{2} + g_{xx}\left(\frac{\partial x}{\partial \varphi}\right)^{2} + g_{yy}\left(\frac{\partial y}{\partial \varphi}\right)^{2} + g_{zz}\left(\frac{\partial z}{\partial \varphi}\right)^{2}. \]

With: \[ \frac{\partial x}{\partial \varphi} = -r\sin\theta\sin\varphi,\qquad \frac{\partial y}{\partial \varphi} = r\sin\theta\cos\varphi,\qquad \frac{\partial z}{\partial \varphi} = 0, \] we obtain: \[ g_{\varphi\varphi} = -\left[ (-r\sin\theta\sin\varphi)^{2} + (r\sin\theta\cos\varphi)^{2} \right]. \]

\[ g_{\varphi\varphi} = -r^{2}\sin^{2}\theta(\sin^{2}\varphi + \cos^{2}\varphi) = -r^{2}\sin^{2}\theta. \]

Result

The transformation from a Cartesian to a spherical metric tensor yields: \[ g_{tt} = 1,\qquad g_{rr} = -1,\qquad g_{\theta\theta} = -r^{2},\qquad g_{\varphi\varphi} = -r^{2}\sin^{2}\theta. \]

In matrix form: \[ g_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -r^{2} & 0 \\ 0 & 0 & 0 & -r^{2}\sin^{2}\theta \end{pmatrix}. \]

This corresponds to the metric of a polar coordinate system in three‑dimensional space.

Conclusion

Applying the metric transformation formula to the transition from Cartesian to spherical coordinates leads to the expected spherical form of the spacetime metric. This exercise illustrates how tensor transformations guarantee the coordinate‑invariance of physical laws within general relativity.

5.9 Further Considerations on Co‑ and Contravariant Transformations

5.9.1 Introduction

In this section we examine how basis vectors and vector components transform under a coordinate transformation. We consider both the direct and inverse transformation and verify their consistency. These considerations form the foundation for understanding covariant and contravariant objects in tensor analysis.

5.9.2 Covariant Transformation of Basis Vectors and Dual Vectors (One‑Forms)

Consider a two‑dimensional vector space with original basis vectors \( e_{1} \) and \( e_{2} \), which are transformed into a new coordinate system with basis vectors \( e_{1}' \) and \( e_{2}' \). This transformation is linear and can be written as: \[ \vec e_{1}' = a_{11} \vec e_{1} + a_{12} \vec e_{2}, \qquad \vec e_{2}' = a_{21} \vec e_{1} + a_{22} \vec e_{2}. \]

In matrix form: \[ \begin{pmatrix} e_{1}' \\ e_{2}' \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix}. \]

Or more compactly: \[ \vec e' = A\,\vec e. \]

5.9.2.1 Inverse Transformation of the Basis Vectors

To find the inverse transformation (from the transformed system back to the original one), we solve for \( e_{1} \) and \( e_{2} \) in terms of \( e_{1}' \) and \( e_{2}' \).

Step 1: Constructing a Linear Combination

We take combinations of the original transformations to isolate \( e_{1} \): \[ a_{22} e_{1}' = a_{11} a_{22} e_{1} + a_{12} a_{22} e_{2}, \] \[ a_{12} e_{2}' = a_{12} a_{21} e_{1} + a_{12} a_{22} e_{2}. \]

Subtracting gives: \[ a_{22} e_{1}' - a_{12} e_{2}' = (a_{11} a_{22} - a_{12} a_{21})\, e_{1}. \]

Thus: \[ e_{1} = \frac{a_{22} e_{1}' - a_{12} e_{2}'}{a_{11} a_{22} - a_{12} a_{21}}. \]

From step 1 we found: \[ e_{1} = \frac{a_{22} e_{1}' - a_{12} e_{2}'}{a_{11}a_{22} - a_{12}a_{21}}. \]

Step 2: Same for \( e_{2} \)

Similarly we find: \[ a_{21} e_{1}' = a_{11}a_{21} e_{1} + a_{12}a_{21} e_{2}, \] \[ a_{11} e_{2}' = a_{11}a_{21} e_{1} + a_{11}a_{22} e_{2}. \]

Multiplying the first equation by \(a_{22}\) and the second by \(a_{12}\), then subtracting: \[ a_{21} e_{1}' - a_{11} e_{2}' = (a_{12}a_{21} - a_{11}a_{22})\, e_{2}. \]

Thus: \[ e_{2} = \frac{-a_{21} e_{1}' + a_{11} e_{2}'}{a_{11}a_{22} - a_{12}a_{21}}. \]

Inverse Transformation in Matrix Form

\[ \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix} = \frac{1}{a_{11}a_{22} - a_{12}a_{21}} \begin{pmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{pmatrix} \begin{pmatrix} e_{1}' \\ e_{2}' \end{pmatrix}. \]

Or more compactly: \[ e = A^{-1} e'. \]

5.9.2.2 Verification of the Inverse Transformation

We verify that \(A^{-1}A = I\).

The original transformation: \[ \begin{pmatrix} e_{1}' \\ e_{2}' \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix}. \]

Now multiply \(A^{-1}\) by \(A\): \[ A^{-1}A = \frac{1}{a_{11}a_{22} - a_{12}a_{21}} \begin{pmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{pmatrix} \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}. \]

Expanding the product: \[ \frac{1}{\det A} \begin{pmatrix} a_{22}a_{11} - a_{12}a_{21} & a_{22}a_{12} - a_{12}a_{22} \\ -a_{21}a_{11} + a_{11}a_{21} & -a_{21}a_{12} + a_{11}a_{22} \end{pmatrix} = \frac{1}{\det A} \begin{pmatrix} \det A & 0 \\ 0 & \det A \end{pmatrix}. \]

Thus: \[ A^{-1}A = I. \]

\[ \frac{a_{22}a_{11} - a_{21}a_{12}}{a_{11}a_{22} - a_{12}a_{21}} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \]

This simplifies to the identity matrix: \[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \]

Thus: \[ A^{-1}A = I. \] The inverse transformation is correct. Q.E.D.

5.9.2.3 Conclusion

We have derived the covariant transformation for basis vectors and its inverse in a two‑dimensional space. We verified that the transformation and its inverse cancel to the identity matrix, confirming the consistency of the transformation between basis vectors in different coordinate systems. This formal consistency is essential for correctly applying tensor transformations in general relativity.

5.9.3 Contravariant Transformation of Vector Components

In differential geometry it is essential to distinguish how basis vectors (covariant) and how vector components (contravariant) transform under a coordinate transformation. In this section we examine the transformation properties of contravariant vector components in a two‑dimensional space.

Vector Invariance and Component Transformation

A vector \(V\) remains geometrically the same under a coordinate transformation, but its components change. In the original coordinate system we write: \[ \vec V = V^{1} \vec e_{1} + V^{2} \vec e_{2}, \] and in the transformed system: \[ \vec V = V^{1'} \vec e_{1}' + V^{2'} \vec e_{2}'. \]

Since the vector itself is invariant, the components \(V^{i}\) must change when the basis vectors change.

Change of Basis

The new basis vectors are linearly related to the original basis vectors via a matrix \(A\): \[ \begin{pmatrix} e_{1}' \\ e_{2}' \end{pmatrix} = A \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix}. \]

Thus: \[ \begin{pmatrix} e_{1}' \\ e_{2}' \end{pmatrix} = A \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix}, \qquad A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}. \]

The inverse transformation for the basis vectors is: \[ \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix} = A^{-1} \begin{pmatrix} e_{1}' \\ e_{2}' \end{pmatrix}. \]

Derivation of the Contravariant Transformation

We express the basis vectors \(e_{1}\) and \(e_{2}\) in terms of \(e_{1}'\) and \(e_{2}'\): \[ V = V^{1} e_{1} + V^{2} e_{2}. \]

Using the inverse transformation: \[ e_{1} = \frac{a_{22} e_{1}' - a_{12} e_{2}'}{\det A}, \qquad e_{2} = \frac{-a_{21} e_{1}' + a_{11} e_{2}'}{\det A}, \] where \[ \det A = a_{11}a_{22} - a_{12}a_{21}. \]

Substituting into the vector: \[ V = V^{1}\frac{a_{22} e_{1}' - a_{12} e_{2}'}{\det A} + V^{2}\frac{-a_{21} e_{1}' + a_{11} e_{2}'}{\det A}. \]

Rewriting: \[ V = \frac{(a_{22}V^{1} - a_{21}V^{2})\,e_{1}' + (-a_{12}V^{1} + a_{11}V^{2})\,e_{2}'} {\det A}. \]

But we also know: \[ V = V^{1'} e_{1}' + V^{2'} e_{2}'. \]

Thus the component transformations are: \[ V^{1'} = \frac{a_{22}V^{1} - a_{21}V^{2}}{\det A}, \qquad V^{2'} = \frac{-a_{12}V^{1} + a_{11}V^{2}}{\det A}. \]

Matrix Form

\[ \begin{pmatrix} V^{1'} \\ V^{2'} \end{pmatrix} = \frac{1}{\det A} \begin{pmatrix} a_{22} & -a_{21} \\ -a_{12} & a_{11} \end{pmatrix} \begin{pmatrix} V^{1} \\ V^{2} \end{pmatrix}. \]

Thus: \[ V' = (A^{-1})^{T} V. \]

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Inverse Transformation of the Components

We start with the transformed vector components: \[ V = V^{1'} e_{1}' + V^{2'} e_{2}'. \]

Using the direct transformation of the basis vectors: \[ e_{1}' = a_{11} e_{1} + a_{12} e_{2}, \qquad e_{2}' = a_{21} e_{1} + a_{22} e_{2}, \] we obtain: \[ V = V^{1'}(a_{11} e_{1} + a_{12} e_{2}) + V^{2'}(a_{21} e_{1} + a_{22} e_{2}). \]

This gives: \[ V = (a_{11}V^{1'} + a_{21}V^{2'})\,e_{1} + (a_{12}V^{1'} + a_{22}V^{2'})\,e_{2}. \]

But we also know that: \[ V = V^{1} e_{1} + V^{2} e_{2}. \]

From this follow the relations for the original vector components: \[ V^{1} = a_{11}V^{1'} + a_{21}V^{2'}, \qquad V^{2} = a_{12}V^{1'} + a_{22}V^{2'}. \]

Matrix Form

\[ \begin{pmatrix} V^{1} \\ V^{2} \end{pmatrix} = \begin{pmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{pmatrix} \begin{pmatrix} V^{1'} \\ V^{2'} \end{pmatrix}. \]

The basis vectors transform according to: \[ \begin{pmatrix} e_{1}' \\ e_{2}' \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} e_{1} \\ e_{2} \end{pmatrix}. \]

In contrast, the vector components transform in the opposite direction: \[ \begin{pmatrix} V^{1'} \\ V^{2'} \end{pmatrix} = \frac{1}{a_{11}a_{22} - a_{12}a_{21}} \begin{pmatrix} a_{22} & -a_{21} \\ -a_{12} & a_{11} \end{pmatrix} \begin{pmatrix} V^{1} \\ V^{2} \end{pmatrix}. \]

5.9.4 Summary of the Transformations

The key relations are as follows:

1.9 Basis vectors (covariant)

\[ e' = A\,e, \qquad e = A^{-1} e'. \]

1.10 Vector components (contravariant)

\[ V' = (A^{-1})^{T} V, \qquad V = A^{T} V'. \]

Thus, if the basis vectors (covariant objects) transform as: \[ e' = A e, \] then the vector components (contravariant objects) transform as: \[ V' = (A^{-1})^{T} V. \]

\[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}. \]

Meanwhile, the relation between the components \(\begin{pmatrix}V^{1} \\ V^{2}\end{pmatrix}\) and \(\begin{pmatrix}V^{1'} \\ V^{2'}\end{pmatrix}\) is given by the transpose of the matrix: \[ A^{T} = \begin{pmatrix} a_{11} & a_{21} \\ a_{12} & a_{22} \end{pmatrix}. \]

The contravariant vector components transform according to: \[ \begin{pmatrix} V^{1'} \\ V^{2'} \end{pmatrix} = \frac{1}{a_{11}a_{22} - a_{12}a_{21}} \begin{pmatrix} a_{22} & -a_{21} \\ -a_{12} & a_{11} \end{pmatrix} \begin{pmatrix} V^{1} \\ V^{2} \end{pmatrix} = (A^{-1})^{T} V. \]

This is the inverse matrix and its transpose.

Conclusion

When the coordinate system changes, the basis vectors transform according to a matrix \(A\), while the vector components transform with the inverse transpose \((A^{-1})^{T}\). This contravariant transformation ensures that the vector \(V\) itself remains invariant: its representation adapts to the changing basis so that its geometric meaning is preserved.

5.10 Considerations on the Minkowski and Schwarzschild Formulas

5.10.1 Minkowski Space

The Minkowski metric is used within special relativity, in which the effects of gravity and acceleration are neglected. In this context, reference frames move uniformly with constant velocity relative to each other, and the coordinate system used is linear and flat.

Consider a point \(K\) in space‑time with its own coordinate system. In this system, \(K\) is always located at the origin, so only time progresses. The distance in space‑time — the interval — is then given by: \[ s = c\,\tau, \] where \(\tau\) is the proper time.

Here, \( \tau \) is the proper time measured by a clock moving along with \(K\). An observer located elsewhere in space‑time, with another inertial frame that moves relative to \(K\), will observe that \(K\) moves through space. The measured velocity of \(K\) is: \[ v^{2} = \frac{x^{2} + y^{2} + z^{2}}{t^{2}}. \]

The Minkowski metric in four‑dimensional space‑time is then written as: \[ s^{2} = c^{2}t^{2} - x^{2} - y^{2} - z^{2}. \]

For an infinitesimal segment along a worldline: \[ ds^{2} = c^{2}dt^{2} - dx^{2} - dy^{2} - dz^{2}. \] This differential segment is essentially a tangent to the worldline in space‑time. Even if the worldline is curved (as in acceleration or in the presence of gravity), we can approximate it locally as composed of linear segments.

The coordinates \(t, x, y, z\) represent four components of a space‑time vector. In an orthogonal coordinate system (as in Minkowski space), the interval can be computed using a generalized Pythagorean theorem. If we take the time component as imaginary \(ict\), and the spatial components as real, the familiar Minkowski form follows.

General Structure of the Interval

We must recognize that \(t, x, y, z\) have magnitude and direction; they are vectors. Finding the magnitude of \(s\) means adding the four vectors. If this coordinate system is orthogonal, the Pythagorean theorem can be applied to the spatial part. If we treat the time part as complex \(ic\,dt\), and the left‑hand side as \(ds = ic\,d\tau\), then by squaring the coordinates we obtain the Minkowski formula.

In two dimensions we can write: \[ s = a_{1}x_{1} + a_{2}x_{2}. \]

To find the magnitude of \(s\), we compute the inner product of \(s\) with itself: \[ s \cdot s = (a_{1}x_{1} + a_{2}x_{2}) \cdot (a_{1}x_{1} + a_{2}x_{2}), \] which results in: \[ s^{2} = a_{1}^{2}x_{1}^{2} + 2a_{1}a_{2}(x_{1}\cdot x_{2}) + a_{2}^{2}x_{2}^{2}. \]

Generalization to Four Dimensions

In four dimensions we generalize this using the metric tensor \(g_{\mu\nu}\): \[ s^{2} = \sum_{\mu,\nu} g_{\mu\nu}\,x^{\mu} x^{\nu} \]

Or in Einstein notation (sum over repeated indices): \[ s^{2} = g_{\mu\nu} x^{\mu} x^{\nu}. \]

When using a locally orthogonal coordinate system, all products with \( \mu \neq \nu \) vanish. If only an infinitesimally small local region is considered, we use \(dx\) instead of \(x\), and similarly for the other coordinates.

Finally, the expression results in a Minkowski or Schwarzschild form: \[ ds^{2} = c^{2}(dx^{0})^{2} - (dx^{1})^{2} - (dx^{2})^{2} - (dx^{3})^{2}. \]

In general: \[ ds^{2} = g_{00}\,c^{2}(dx^{0})^{2} + g_{11}(dx^{1})^{2} + g_{22}(dx^{2})^{2} + g_{33}(dx^{3})^{2}. \]

In Minkowski space the components of the metric tensor are constant: \[ g_{00} = 1, \qquad g_{11} = g_{22} = g_{33} = -1. \]

Meaning of the Minkowski Formula

The Minkowski interval formula reads: \[ ds^{2} = c^{2}dt^{2} - dx^{2} - dy^{2} - dz^{2} = c^{2}dt'^{2} - dx'^{2} - dy'^{2} - dz'^{2}. \]

The left‑hand side represents an object located in its own (co‑moving) reference frame: it experiences only a progression in its own proper time \( \tau \).

An observer in the coordinate system \(t, x, y, z\) sees the object move with velocity: \[ v^{2} = \frac{dx^{2} + dy^{2} + dz^{2}}{dt^{2}}. \]

A second observer in another inertial frame \(t', x', y', z'\) measures: \[ v'^{2} = \frac{dx'^{2} + dy'^{2} + dz'^{2}}{dt'^{2}}. \]

The relation between proper time \( \tau \) and coordinate time \(t\) is given by: \[ ds^{2} = c^{2}dt^{2} - dx^{2} - dy^{2} - dz^{2} = c^{2}d\tau^{2}. \]

Thus: \[ c^{2}d\tau^{2} = c^{2}dt^{2}\left(1 - \frac{v^{2}}{c^{2}}\right) = c^{2}dt^{2}\,\gamma^{-2}. \]

Here \( \tau \) is the so‑called proper time, the time of a moving clock located at the origin of its own co‑moving coordinate system.

The relation between proper time and observer time is: \[ d\tau^{2} = \frac{dt^{2}}{\gamma^{2}}, \qquad dt = \gamma\,d\tau, \] with the Lorentz factor: \[ \gamma = \frac{1}{\sqrt{1 - v^{2}/c^{2}}}. \]

Because \( \gamma \ge 1 \), we have \( d\tau \le dt \): a moving clock runs slower than a clock at rest from the perspective of an external observer.

5.10.2 Transformations Performed by Schwarzschild

The Schwarzschild metric extends the Minkowski metric by taking into account the effects of mass and gravity. Unlike the flat space‑time of special relativity, this results in curved space‑time. This curvature manifests itself in a non‑linear coordinate system adapted to the spherical symmetry around a massive body.

From Cartesian to Spherical Coordinates

Schwarzschild begins with the usual flat (Cartesian) coordinates and performs a transformation to spherical coordinates \(r, \theta, \phi\). This results in the following expression for the space‑time interval (in natural units \(G=c=1\), though we keep \(c\) explicit here): \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dr^{2}}{\sigma^{2}} - r^{2} d\theta^{2} - r^{2}\sin^{2}\theta\, d\phi^{2}, \] with: \[ \sigma^{2} = 1 - \frac{2GM}{rc^{2}}. \]

The Determinant of the Metric

The determinant \(g\) of the metric tensor in these coordinates is: \[ g = \sigma^{2} \cdot \left(-\frac{1}{\sigma^{2}}\right) \cdot (-r^{2}) \cdot (-r^{2}\sin^{2}\theta) = -r^{4}\sin^{2}\theta. \]

Einstein preferred that in suitable coordinates the determinant satisfy \(g = -1\) (as in the Minkowski metric). Schwarzschild therefore investigates whether a coordinate transformation exists that yields this condition.

Transformation to New Coordinates

(Next step: here you may introduce the Schwarzschild radial transformation, such as \(R^{3} = r^{3} + \alpha^{3}\), or isotropic coordinates, depending on the direction you wish to develop.)

To normalize the determinant to \( g = -1 \), Schwarzschild defines new coordinates \(x_{1}, x_{2}, x_{3}\), based on: \[ \frac{dr}{dx_{1}} = \frac{1}{r^{2}}, \qquad \frac{d\theta}{dx_{2}} = \frac{1}{\sin\theta}, \qquad \frac{d\phi}{dx_{3}} = 1. \]

Schwarzschild notes: “The new variables are the polar coordinates with determinant 1.” To obtain these derivatives, he finds the following relations: \[ x_{1} = \frac{r^{3}}{3}, \qquad x_{2} = -\cos\theta, \qquad x_{3} = \phi. \]

These new coordinates transform the metric into: \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dx_{1}^{2}}{r^{4}\sigma^{2}} - r^{2}\cdot \frac{1}{\sin^{2}\theta}\,dx_{2}^{2} - r^{2}\sin^{2}\theta\,dx_{3}^{2}. \]

Substitution of Trigonometric Relations

Because \(x_{2} = -\cos\theta\), we have: \[ x_{2}^{2} = \cos^{2}\theta = 1 - \sin^{2}\theta \quad\Rightarrow\quad \sin^{2}\theta = 1 - x_{2}^{2}. \]

Thus we rewrite the metric as: \[ ds^{2} = \sigma^{2} c^{2} dt^{2} - \frac{dx_{1}^{2}}{r^{4}\sigma^{2}} - \frac{r^{2}}{1 - x_{2}^{2}}\,dx_{2}^{2} - r^{2}(1 - x_{2}^{2})\,dx_{3}^{2}. \]

The New Metric Components

The components of the metric tensor \(g_{\mu\nu}\) in these transformed coordinates are now: \[ g_{00} = \sigma^{2}, \qquad g_{11} = -\frac{1}{r^{4}\sigma^{2}}, \qquad g_{22} = -\frac{r^{2}}{1 - x_{2}^{2}}, \qquad g_{33} = -r^{2}(1 - x_{2}^{2}). \]

The determinant \(g\) of this tensor is now: \[ g = g_{00}\,g_{11}\,g_{22}\,g_{33} = -1. \]

Exactly as desired. The transformation performed by Schwarzschild is therefore valid and results in a metric with determinant \(-1\), despite the curved nature of space‑time.

Special Cases

In the specific case \( \theta = 90^\circ \), we have \( \cos\theta = 0 \) and thus \( x_{2} = 0 \), which leads to: \[ \sin^{2}\theta = 1 \quad\Longrightarrow\quad ds^{2} = \sigma^{2}c^{2}dt^{2} - \frac{dx_{1}^{2}}{r^{4}\sigma^{2}} - r^{2}dx_{2}^{2} - r^{2}dx_{3}^{2}. \]

In this plane around the equatorial region, the metric becomes even simpler in form.

5.11 Schwarzschild’s: “On the Gravitational Field of a Mass Point According to Einstein’s Theory”

Karl Schwarzschild’s goal in his 1916 paper was to find an exact solution of the Einstein field equations in vacuum. This solution describes the space‑time around a point mass moving along a geodesic in a four‑dimensional manifold, with the space‑time interval \(ds\) playing a central role.

Conditions for the Solution

The following conditions are imposed on the solution:

1. Time independence: All components of the metric are independent of the time coordinate \(x^{4}\).
2. No space‑time coupling: The mixed components \(g_{\rho 4} = g_{4\rho} = 0\) for \(\rho = 1,2,3\).
3. Spherical symmetry: The solution is invariant under orthogonal transformations (rotations) of \(x_{1}, x_{2}, x_{3}\); this reflects spherical symmetry.
4. Asymptotic flatness: At infinite distance the components of the metric tensor approach: \[ \lim_{r\to\infty} g_{44} = 1, \qquad \lim_{r\to\infty} g_{11} = g_{22} = g_{33} = -1. \]

From Cartesian to Spherical Coordinates

Schwarzschild begins with a general metric in Cartesian coordinates: \[ ds^{2} = F\,dt^{2} - G\,(dx^{2} + dy^{2} + dz^{2}) - H\,(x\,dx + y\,dy + z\,dz)^{2}. \]

He then performs the standard transformations to spherical coordinates: \[ x = r\sin\vartheta\cos\phi,\qquad y = r\sin\vartheta\sin\phi,\qquad z = r\cos\vartheta. \]

After substitution, the space–time interval in spherical coordinates becomes: \[ ds^{2} = F\,dt^{2} - G\,(dr^{2} + r^{2}d\vartheta^{2} + r^{2}\sin^{2}\vartheta\,d\phi^{2}) - H\,r^{2}dr^{2}. \]

This can be rewritten as: \[ ds^{2} = F\,dt^{2} - \left(G + Hr^{2}\right)dr^{2} - G r^{2}\left(d\vartheta^{2} + \sin^{2}\vartheta\,d\phi^{2}\right). \]

Transformation to Determinant 1

Because the determinant of the metric in this form is not equal to \(-1\), Schwarzschild performs a transformation to new variables that do satisfy this condition. He defines: \[ x_{1} = \frac{r^{3}}{3}, \qquad x_{2} = -\cos\vartheta, \qquad x_{3} = \phi. \]

With these definitions, the line element becomes: \[ ds^{2} = F\,dt^{2} - \left(\frac{G}{r^{4}} +\frac{ H}{r^{2}}\right)\,dx_{1}^{2} -Gr^2\left[ \frac{dx_{2}^{2}}{1 - x_{2}^{2}} + (1 - x_{2}^{2})\,dx_{3}^{2}\right] \]

The Schwarzschild Solution

By inserting this metric into the Einstein field equations and solving them in vacuum (\(T_{\mu\nu} = 0\)), Schwarzschild finds the well‑known solution: \[ ds^{2} = c^{2}d\tau^{2} = \left(1 - \frac{2GM}{c^{2}r}\right)c^{2}dt^{2} - \left(1 - \frac{2GM}{c^{2}r}\right)^{-1}dr^{2} - r^{2}d\theta^{2} - r^{2}\sin^{2}\theta\,d\phi^{2}. \tag{12} \]

This expression describes the curved space–time surrounding a spherically symmetric mass point in vacuum. Although Schwarzschild began his derivation in Cartesian coordinates, the final solution is more convenient and insightful in spherical coordinates, given the spherical symmetry of the problem.

Schwarzschild Solution in Cartesian Coordinates

There also exists a less common form of the Schwarzschild solution in Cartesian coordinates, which reads: \[ ds^{2} = c^{2}d\tau^{2} = \left(1 - \frac{2GM}{c^{2}r}\right)c^{2}dt^{2} - (dx^{2} + dy^{2} + dz^{2}) - \frac{2GM}{c^{2}r\left(1 - \frac{2GM}{c^{2}r}\right)} \left(\frac{x\,dx + y\,dy + z\,dz}{r}\right)^{2}. \tag{13} \]

However, this form is rarely practical, because spherical coordinates align much better with the symmetry of the problem — for example in applications such as describing black holes or the exterior of stars.

Sources

• K. Schwarzschild, On the Gravitational Field of a Point‑Mass, According to Einstein's Theory, 13 January 1916.
• G. Oas, various discussions and analyses of the Schwarzschild solution. See also the Bibliography chapter at the end of this document.

The Schwarzschild solution forms a cornerstone of general relativity and is widely used in astrophysics in the study of black holes, neutron stars, and other objects with extremely strong gravitational fields.