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Tensors and Relativity: Assignment 1

# Problem

### Peter Dunsby Room 310a Applied Mathematics

• (1) Show that the electromagnetic wave equation,

is invariant under a Lorentz transformation.

• (2) The equation for a spherical pulse of light starting from the origin at is,

Show from the Lorentz transformations that observer will also measure this same pulse to be spherical, in accord with Einstein's second postulate stating that the velocity of light is the same for all observers.

• (3) A -meson with an average lifetime of sec is created in the upper atmosphere at an altitude of 6000 m. When it is created it has a velocity of in a direction towards the earth. What is the average distance that it will travel before decaying, as determined by an observer on earth?

Consider an observer at rest with respect to the -meson. What is the distance he measures from the point of creation of the -meson to the earth? Comment on the physical significance of these results.

• (4) Consider a radioactive nucleus that moves with a constant speed relative to the laboratory. The nucleus decays and emits an electron with a speed in a direction perpendicular to the direction of (the laboratory's) motion as determined by an observer at rest with respect to the nucleus. Find the velocity and the direction of motion of the electron as measured by an observer in the laboratory frame.
• (5) The speed of light in still water is , where the index of refraction for water is approximately . Fizeau, in 1851, found that the speed of light in the water moving with a speed V (relative to the laboratory) could be expressed as

where the ``dragging coeficient'' was measured by him to be . determine the value of k predicted by the Lorentz velocity transformations.

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