5.1 Derivation and application of Lorentz transformations

Lorentz transformations are the mathematical heart of special relativity. They describe how space and time coordinates change between different reference frames moving at constant velocities relative to each other.

These transformations lead to mind-bending effects like time dilation and length contraction. They also reveal the relativity of simultaneity, showing that the order of events can differ depending on your frame of reference.

Lorentz Transformations

Lorentz Factor and Relativistic Effects

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• Lorentz factor ($\gamma$) depends on the relative velocity between two reference frames
  • Defined as $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$, where $v$ is the relative velocity and $c$ is the speed of light
  • Approaches infinity as the relative velocity approaches the speed of light
• Time dilation occurs when an observer in one reference frame measures a longer time interval than an observer in another reference frame
  • Proper time ($\tau$) is the time measured by a clock in its own rest frame
  • Dilated time ($t$) is related to proper time by $t = \gamma \tau$
  • Example: Muons created in Earth's upper atmosphere have a longer lifetime from Earth's reference frame due to time dilation
• Length contraction happens when an object appears shorter along the direction of motion to an observer in a different reference frame
  • Proper length ($L_0$) is the length of an object in its own rest frame
  • Contracted length ($L$) is related to proper length by $L = \frac{L_0}{\gamma}$
  • Example: A spacecraft traveling at high velocity would appear shorter to a stationary observer

Relativity of Simultaneity

• Relativity of simultaneity states that events that are simultaneous in one reference frame may not be simultaneous in another
• The order of events can differ between reference frames if the events are separated by a space-like interval
  • Space-like interval: $(\Delta s)^2 = (\Delta x)^2 - c^2(\Delta t)^2 > 0$
• Example: Two lightning strikes equidistant from an observer on a moving train may appear simultaneous to the observer on the train but not to an observer on the platform

Proper Measurements

Proper Time and Length

• Proper time ($\tau$) is the time measured by a clock between two events that occur at the same location in the clock's rest frame
  • Proper time is always the shortest time interval between two events as measured by any clock
• Proper length ($L_0$) is the length of an object as measured in its own rest frame
  • Proper length is always the longest length of an object as measured by any observer

Invariant Interval

• The invariant interval ($\Delta s$) is a combination of the spatial and temporal separations between two events that remains constant in all reference frames
  • Defined as $(\Delta s)^2 = c^2(\Delta t)^2 - (\Delta x)^2$
  • Can be classified as space-like ($(\Delta s)^2 > 0$), time-like ($(\Delta s)^2 < 0$), or light-like ($(\Delta s)^2 = 0$)
• The proper time between two events is related to the invariant interval by $\tau = \frac{\sqrt{-(\Delta s)^2}}{c}$ for time-like intervals
• Example: The invariant interval between two events on a photon's world line is always zero (light-like interval)

Relativistic Velocity

Velocity Addition Formula

• The relativistic velocity addition formula combines velocities in different reference frames
  • For velocities $u$ and $v$ in the $x$-direction, the combined velocity $w$ is given by $w = \frac{u + v}{1 + \frac{uv}{c^2}}$
  • Velocities do not simply add linearly as they do in classical mechanics
• The formula ensures that the combined velocity never exceeds the speed of light
  • As one velocity approaches $c$, the combined velocity approaches $c$ regardless of the other velocity
• Example: If a spacecraft is traveling at 0.8$c$ relative to Earth and fires a missile at 0.6$c$ relative to the spacecraft, the missile's velocity relative to Earth is $\frac{0.8c + 0.6c}{1 + \frac{(0.8c)(0.6c)}{c^2}} \approx 0.94c$

Rapidity

• Rapidity ($\phi$) is a measure of the relative velocity between two reference frames that simplifies Lorentz transformations
  • Defined as $\phi = \tanh^{-1}(\frac{v}{c})$, where $v$ is the relative velocity and $c$ is the speed of light
  • Lorentz factor can be expressed as $\gamma = \cosh(\phi)$
• Rapidities add linearly, unlike velocities
  • For two reference frames with rapidities $\phi_1$ and $\phi_2$, the combined rapidity is simply $\phi_1 + \phi_2$
• Example: If frame B has a rapidity of 0.8 relative to frame A, and frame C has a rapidity of 0.6 relative to frame B, then frame C has a rapidity of 1.4 relative to frame A