# Concepts through Pictures

Who discovered $E=\Delta mc^2$? Even Einstein admitted that Poincaré derived the relation in 1900 by analyzing the recoil of an artillery canon---five years before he did.

Is $S\propto A\propto M^2$ the entropy of a black hole where $A$ is the area of the event horizon and $M$ is its mass, implying that $T\propto M^{-1}$ be its temperature? There is no physical system, or any final state of one, that would admit such a thermodynamic description for it would violate the same law it claims to have derived---the second law.

Are there gravitational waves, and if there are, why should they propagate at the speed of light? The general theory evolved as a generalization of the Minkowski metric which claims to describe inertial systems. The generalization would constitute systems which accelerate, but all forms of acceleration are not equivalent. Consequently, there is no reason why gravitational waves should propagate at the speed of light, and that far from gravitating bodies the metric should reduce to the Minkowski metric.

Are the curvilinear triangles that make up the tessellation different in size? No, Poincaré discovered a concept of length under which all the cells of the tessellation are of equal size, and so introduced hyperbolic geometry into mainstream mathematics---and even relativity!

Has anyone seen bodies contract in the direction of motion? The contraction hypothesis was used to explain the lack of time differences in the two journeys, parallel and perpendicular to the direction of the motion of the aether, in the Michelson-Morley experiment based on an incorrect emission theory.

Does the Sagnac effect contradict relativity? The Sagnac effect was used as an experimental proof that there is a time difference for light to cover a circular distance in the direction of rotation of the disc and in the opposite direction---giving rise to an interference pattern and a shift in the fringes.

Can the acceleration of a rocket ship nullify the force of gravity? Gamow's sketch of Einstein's gendanken experiment, showing the equivalence between acceleration and gravity, cannot be realized because gravitational and centrifugal forces are different---in the optical analogy, gravitation would influence the index of refraction whereas the centrifugal potential is built into the metric.

The "twin paradox" is used to show that a clock in motion goes slower than a stationary clock so that an astronaut who makes a round trip in space will have aged less than his twin brother who stayed at home. The clocks are in uniform motion so that either clock can be taken to be the "moving" one. The gedanken experiment assumes that acceleration and deceleration have no effect, but without them the astronaut could not be propelled into space and complete a round trip.