The Biggest Ideas in the Universe: Space, Time, and Motion

there is an alternative view: that space isn’t a thing at all, it’s just a way to repackage the fact that any two objects are characterized by a quantity called “the distance between them.” From this perspective, space is fundamentally relational.

“spacetime tells matter how to move; matter tells spacetime how to curve.”

< 1, where r is a radial coordinate measured from the origin.

Since the day Isaac Newton laid down his laws, a number of other kinds of laws have been suggested for fundamental physical systems. James Clerk Maxwell wrote down a set of equations for electricity and magnetism; Albert Einstein proposed an equation for the curvature of spacetime; Erwin Schrödinger suggested an equation for the wave function of a

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this failure of the two paths to give us the same results provides a nice way to formalize what we mean by “curvature.”

Today, most physicists would come down on the side of Newton, treating space as a thing in itself, for a couple of reasons. First, the space in between objects is not empty; it is filled with fields of various sorts. Second, space (as part of spacetime) has a life of its own; Einstein showed that the geometry of space responds to energy and can cha

Any index that is summed over is called a dummy index, while one that is not summed over is a free index.

and also under shifts in time