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How do Lorentz transformations keep the speed of light the same for different observers?

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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.

Lorentz transformations ensure the speed of light remains constant for all observers by replacing the classical assumption of absolute space and time with a framework where these dimensions depend on the observer's state of motion.12

In classical physics, Galilean transformations assumed time was identical for all observers, which caused the calculated speed of light to vary depending on the observer's frame of reference—a result contradicted by experimental evidence. To resolve this, Albert Einstein adopted Lorentz transformations, which mathematically link the space and time coordinates of different inertial frames of reference.32

These transformations account for the relativity of simultaneity, meaning that as an observer's speed changes, their measurements of length and time intervals adjust—a phenomenon known as length contraction and time dilation. By incorporating these changes, the Lorentz equations guarantee that the speed of light is measured as $c$ in all inertial frames, effectively replacing the classical law of velocity addition with a relativistic version that prevents the speed of light from being exceeded or altered by the observer's relative motion.23