Show that the spacetime interval given by Eq. 15.18 is...

Show that the spacetime interval given by Eq. 15.18 is invariant with respect to the Lorentz transformation. That is, show that $(d s)^{2}=\left(d s^{\prime}\right)^{2},$ where $\left(d s^{\prime}\right)^{2}=$ $\left(c d r^{\prime}\right)^{2}-\left(d x^{\prime}\right)^{2}$

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Key Concepts

Lorentz Transformation

Lorentz transformations are the set of linear transformations that relate the space and time coordinates of events as measured in different inertial frames moving at constant velocity relative to one another. They ensure that the speed of light remains constant in all inertial frames and preserve the form of the spacetime interval (i.e., the Minkowski metric). This preservation, reflected by the equality of ds² in all inertial frames, is a cornerstone of the theory of special relativity.

Spacetime Interval

The spacetime interval is a fundamental measure in the geometry of Minkowski spacetime, defined by a combination of temporal and spatial separations between two events. It is given by an expression such as ds² = c² dt² - dx² for simplified cases, and it encodes the invariant separation that remains constant across different inertial frames. This invariance underpins the causal structure of special relativity, distinguishing between timelike, spacelike, and lightlike separations.

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