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

The distance traveled is just the area of the rectangle stretching horizontally from 0 to Δt and vertically from 0 to v.

Same geometry, but different versions of the line element, because we’re in different coordinate systems.

In relativity, all of these ideas are bundled up into the energy-momentum tensor (also known as the stress-energy tensor), usually written Tµν.

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

The “Hamiltonian” itself is simply the energy of the system, expressed as a function of position and momentum (and not as a function of velocity or other derivatives).

The main trick is to define the four-velocity as a derivative with respect to the proper time τ along the trajectory, rather than with respect to the coordinate time t.

These two questions—what is the instantaneous rate of change of some quantity, and how do we accumulate quantities over time?—are precisely the subject of calculus.

Let be a “unit vector”—a

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