Question

Find the geodesics of the metric \(d s^{2}=d x^{2}+x^{2} d y^{2}\).

Short answer

The geodesics for the given metric are the set of curves defined by \(x = a \ cosh(bs)\) and \(y = (1/b) \ sinh(bs) + c\), where a, b and c are constants determined by the initial conditions.

Step-by-step solution

Write down the metric in a different coordinate system

The given metric is already in polar coordinates, so it can be used directly. The initial metric is \(d s^{2}=d x^{2}+x^{2} d y^{2}\).

Write down the geodesic equations

The geodesic equations are derived from the metric. They are given by: \(\ddot{x} - x\dot{y}^{2} = 0 \) and \(\ddot{y} + 2\frac{\dot{x}}{x} \dot{y} = 0\), where the dots denote derivatives with respect to a curve parameter, usually chosen to be the arc length s. A dot over a symbol means differentiation with respect to this curve parameter.

Solve the geodesic equations

Solving the above system of second-order differential equations, we find that the solutions (geodesic equations) are: \(x = a \ cosh(b s)\) and \(y = (1/b) \ sinh(b s) + c\), where a, b and c are constants to be determined by initial conditions.

Key concepts

Differential Geometry

Differential geometry is an expansive field in mathematics that uses the techniques of calculus and algebra to study problems in geometry. Above all, it concerns the properties and behavior of curves and surfaces in space — these are the objects that geodesic equations are typically concerned with. Central to differential geometry are concepts such as manifolds, which generalize the idea of curves and surfaces to higher dimensions, and the curvature, which quantifies how an object deviates from being flat.

For instance, in the example of finding geodesics of the metric given by the expression \(d s^{2}=d x^{2}+x^{2} d y^{2}\), differential geometry provides the framework for understanding this metric as a way to measure distances on a surface. These distances are not like our everyday Euclidean distances; instead, they take into account the 'shape' of the space. This is particularly useful in the general theory of relativity, where the 'shape' of spacetime becomes an important factor in describing gravitational phenomena.

Identifying geodesics, which are the 'straightest' possible paths between points on a curved space, is an essential problem in differential geometry. These paths minimize or maximize a certain quantity, typically the length or energy, just like straight lines do in flat geometry. The geodesic equations derived from the given metric \(d s^{2}\) represent those paths mathematically.

Metrics in General Relativity

In the context of general relativity, metrics take on a pivotal role. General relativity, conceived by Albert Einstein, depicts gravity not as a force but as a result of the curvature of spacetime, created by mass and energy. Metrics, like the one in the given exercise \(d s^{2}=d x^{2}+x^{2} d y^{2}\), describe this curvature precisely and provide a way to calculate distances and times near massive objects.

In simpler terms, metrics explain how spacetime gets 'stretched' or 'squeezed' by mass and energy. These deformations affect the movements of objects and the flow of time — phenomena that can be described by the geodesic equations derived from the metric. Geodesic motions represent the trajectories that free-falling objects follow, influenced solely by the curvature of spacetime, without any other forces acting upon them.

The geodesic equations from the exercise are simplified versions of what we find in general relativity, but they embody the same principle: they characterize the most 'natural' paths through a curved space. Solving such equations yields valuable insights into the structure of space itself and the potential paths an object may traverse within that space.

Second-order Differential Equations

The geodesic equations that stem from the metric equation in our example, such as \(\ddot{x} - x\dot{y}^{2} = 0\) and \(\ddot{y} + 2\frac{\dot{x}}{x} \dot{y} = 0\), are second-order differential equations. These are equations that involve the second derivatives of a function, indicating how the rate of change of a quantity is itself changing. In physical systems, second-order differential equations frequently appear in contexts ranging from simple harmonic motion to the equations of motion under general relativity.

The process of solving these equations often requires integrating twice, as each integration reduces the order of the derivative by one. The solutions to these equations, like \(x = a \cosh(b s)\) and \(y = (1/b) \sinh(b s) + c\) in our exercise, are functions that describe the behavior of a system; in this case, the geodesic paths. What's crucial about second-order differential equations is that they typically allow for two constants of integration, here represented by \(a\), \(b\), and \(c\), which are determined by initial conditions. These constants make it possible to adapt the general solution to specific circumstances, enabling precision in the modeling of physical phenomena.