Length Contraction

\(L = \frac{L_0}{\gamma} = L_0 \sqrt{1 - \frac{v^2}{c^2}}\)

Variables:

Description/Usage:

Objects are measured to be shorter along the direction of motion relative to an observer for whom they are moving. For instance, a fast-moving rod or spaceship appears foreshortened. Like time dilation, this effect is only significant at speeds close to \(c\).

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Mass–Energy Equivalence

\(E = m c^2\)

Variables:

Description/Usage:

Shows mass is a concentrated form of energy. A small amount of mass can convert to a huge amount of energy (because \(c^2\) is large). This underlies nuclear reactions where binding energy differences (mass defects) release energy.

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Relativistic Momentum

\(\displaystyle p = \gamma m v = \frac{m v}{\sqrt{1 - v^2/c^2}}\)

Variables:

Description/Usage:

Mathematical concept visualization

Schwarzschild Radius (Black Hole)

\(r_s = \frac{2 G M}{c^2}\)

Variables:

\(M = mass of a (spherically symmetric, non-rotating) object, G = gravitational constant, c = speed of light, r_s = Schwarzschild radius.\)

Description/Usage:

If an object of mass \(M\) is compressed within this radius, its escape velocity exceeds \(c\) and it becomes a black hole. For example, the Earth’s \(r_s\) is ~9 mm, the Sun’s ~3 km. This formula comes from General Relativity’s solution for spacetime around a mass.

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Gravitational Time Dilation (weak field)

\(t_f = t_p \sqrt{1 - \frac{2GM}{r c^2}}\)

Variables:

\(t_p is proper time (time measured by a clock at radius r in gravitational field), t_f is time measured by a far-away observer, M is mass of gravitating body, r is radial position of the clock (from center of mass).\)

Description/Usage:

Clocks deeper in a gravitational well (closer to a mass) run slower relative to clocks far away. This is a prediction of GR (even the GPS satellites must correct for this and special relativistic effects to synchronize with Earth clocks). When \(r\) is very large (far away), the factor tends to 1 (no dilation).

Mathematical concept visualization