Problem 37 - A Course in Mathematical Methods for Physicists

Question

The line element in terms of the metric tensor, \(g_{\alpha \beta}\) is given by $$ d s^{2}=g_{\alpha \beta} d x^{\alpha} d x^{\beta}. $$ Show that the transformed metric for the transformation \(x^{\prime \alpha}=x^{\prime \alpha}\left(x^{\beta}\right)\) is given by $$ g_{\gamma \delta}^{\prime}=g_{\alpha \beta} \frac{\partial x^{\alpha}}{\partial x^{\prime} \gamma} \frac{\partial x^{\beta}}{\partial x^{\prime \delta}} $$

Step-by-Step Solution

Answer

The transformed metric for the transformation \(x^{\prime\alpha}=x^{\prime \alpha}\left(x^{\beta}\right)\) is given by \(g_{\gamma \delta}^{\prime}=g_{\alpha \beta} \frac{\partialx^{\alpha}}{\partial x^{\prime} \gamma} \frac{\partial x^{\beta}}{\partialx^{\prime \delta}}\).

Step 1: Establish the connection between \(d s^{2}\) in old and new coordinates

We establish the connection between the line element \(d s^{2}\) in the old and new coordinates. The line element in terms of the transformed coordinates is given by \(d s^{\prime2} = g_{\gamma \delta}^{\prime} d x^{\prime \gamma} d x^{\prime \delta}\), where \(g_{\gamma \delta}^{\prime}\) is the transformed metric tensor and \(d x^{\prime \gamma}\) and \(d x^{\prime \delta}\) are the differentials of the transformed coordinates.

Step 2: Express the differential of the transformed coordinates in terms of original coordinates

We express the differential of the transformed coordinates in terms of the differentials of the original coordinates: \(d x^{\prime \gamma} = \frac{\partial x^{\prime \gamma}}{\partial x^{\alpha}} d x^{\alpha}\) and \(d x^{\prime \delta} = \frac{\partial x^{\prime \delta}}{\partial x^{\beta}} d x^{\beta}\). Replace these differentials in the previous equation for \(d s^{\prime 2}\). We get \(d s^{\prime 2} = g_{\gamma \delta}^{\prime} \frac{\partial x^{\prime \gamma}}{\partial x^{\alpha}} \frac{\partial x^{\prime \delta}}{\partial x^{\beta}} d x^{\alpha} d x^{\beta}\)

Step 3: Compare with the original line element expression

By comparing this with the original line element expression \(d s^{2}=g_{\alpha \beta} d x^{\alpha} d x^{\beta}\), we find that the coefficients of \(d x^{\alpha} d x^{\beta}\) must be equal in both expressions. This leads us to the conclusion that \(g_{\gamma \delta}^{\prime} \frac{\partial x^{\prime \gamma}}{\partial x^{\alpha}} \frac{\partial x^{\prime \delta}}{\partial x^{\beta}} = g_{\alpha \beta}\). Hence, we find the transformed metric tensor in the new coordinates as \(g_{\gamma \delta}^{\prime}=g_{\alpha \beta} \frac{\partialx^{\alpha}}{\partial x^{\prime} \gamma} \frac{\partial x^{\beta}}{\partialx^{\prime \delta}}\) as required.

Key Concepts

Coordinate Transformation

The concept of coordinate transformation is foundational in understanding mathematical spaces in physics and engineering. Essentially, it involves changing the viewpoint from which you describe your space, which could be a physical location or a more abstract phenomenon. In technical terms, it's a way to map one set of coordinates onto another, usually to simplify calculations or align them with physical reality.

Let's consider an analogy: if you move from one city to another, you may have to adapt to a different layout of streets. Similarly, in coordinate transformation, the 'layout' of your mathematical 'city' changes. This is critical in areas such as relativity theory, where observer-dependent descriptions of space-time require careful transformation between different reference frames.

Improving upon our existing solution, let's visualize this transformation as a function: \(x^{\prime\alpha} = x^{\prime\alpha}(x^{\beta})\). This denotes that every point in our space described by coordinates \(x^{\alpha}\) can be translated to a new set of coordinates \(x^{\prime\alpha}\). Such transformations are essential to ensure that physical laws are consistent across different perspectives, which is a core principle in physics known as covariance.

Differential Geometry

In the realm of differential geometry, we deal with curves, surfaces, and more complex structures known as manifolds. Imagine drawing shapes on a balloon. As the balloon inflates or deflates, the shapes stretch and compress but still retain their fundamental properties. Differential geometry studies these properties under various transformations.

The key concept here is curvature, the way a geometric object deviates from being flat, like the surface of a sphere compared to a sheet of paper. Now apply this idea to the universe's fabric, and you have a glimpse of how differential geometry underpins Einstein's General Theory of Relativity, describing gravity as a result of space-time curvature.

Incorporating this into our exercise, we examine how the metric tensor, which quantifies distances and angles in a space, behaves under coordinate transformation. To solidify understanding, always remember that differential geometry helps us describe how objects bend and twist in space—imagine the warping of a space-time map near a massive object like a black hole.

Line Element

The line element serves as a fundamental concept in the study of space and represents the square of an infinitesimally small distance between two nearby points in a space. Think of it as a tiny ruler that measures how the space stretches or squashes at every point.

Mathematically, it's described as \(ds^{2} = g_{\alpha \beta} dx^{\alpha} dx^{\beta}\), where \(g_{\alpha \beta}\) is the metric tensor. In physics, especially in Einstein's theory of relativity, the line element helps describe the geometry of space-time, encompassing both spatial distances and time intervals.

When we apply a coordinate transformation, the line element helps us ensure that the physical distances we're measuring are consistent across different coordinate systems. This is tantamount to us measuring a table with a ruler and ensuring that the length does not change whether we measure it in inches or centimeters. The line element thus acts as a bridge between different descriptions of space.

Tensor Calculus

Delving into the universe of tensor calculus, we encounter mathematical objects that generalize the concepts of scalars, vectors, and matrices. Tensors can be understood as multi-dimensional arrays of numbers that transform in a particular way under coordinate transformations.

Why use tensors? Because they encapsulate the laws of physics in a form that remains constant across all coordinate systems—a property known as invariance. For instance, in our exercise, we dealt with the metric tensor \(g_{\alpha \beta}\), which provides a way to measure lengths and angles in various geometries.

Understanding tensor calculus unlocks the ability to work with and describe physical laws in any coordinate system. The process of transforming tensors reveals how quantities such as energy, momentum, and force change when you alter your frame of reference—the view from which you're observing a physical system. This is crucial in fields like continuum mechanics and the theory of relativity, where we deal with diverse physical frames.