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Why do all inertial observers agree on the space-time interval between two events?

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Brian Greene sits in a living room, holding a pen, with handwritten equations m = E/c^2 and E = mc^2 on a whiteboard beside him, highlighting c^2 as the speed of light. — relativity Brian Greene kicking off his *Your Daily Equation* video series with Albert Einstein's famous equation E = mc².

In special relativity, inertial observers do not agree on the individual spatial distance or time interval between two events, as these measurements depend on the observer's frame of reference. However, they all agree on a specific algebraic combination of these values known as the spatiotemporal interval.12

This invariance exists because the laws of physics, specifically the constant velocity of light in empty space, require that space and time be treated as a single, unified four-dimensional space-time continuum. The interval, often denoted as $d s^{2} = c^{2} d t^{2} - d x^{2} - d y^{2} - d z^{2}$ , remains constant when transforming between different inertial frames using Lorentz transformations.234

By ensuring this quantity is invariant, the theory accounts for the fact that while simultaneity and distance are relative to one's perspective, the underlying geometric structure of the universe remains consistent for all non-accelerating observers.23