Lorentz transformation

Lorentz boost x direction standard configuration

Lorentz transformation refers to a set of equations in special relativity that describe how, according to the theory of special relativity, the time and position of an event change when observed from different inertial frames of reference. These transformations are named after the Dutch physicist Hendrik Lorentz who proposed them in 1904, before Albert Einstein's paper on special relativity was published.

Overview

The Lorentz transformations are derived from two postulates proposed by Einstein:

1. The laws of physics are invariant (identical) in all inertial frames of reference (i.e., frames of reference with no acceleration).
2. The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer.

These transformations mathematically formulate how the measurements of time, length, and mass of objects change as these objects move relative to an observer. The transformations are crucial for understanding phenomena such as time dilation, length contraction, and the relativity of simultaneity, which are all counterintuitive effects predicted by special relativity.

Mathematical Formulation

The Lorentz transformation equations for the time and space coordinates of two inertial frames of reference (commonly labeled as S and S') in standard configuration (where S' moves at a velocity v along the x-axis of S) are:

x' = γ(x - vt)

y' = y

z' = z

t' = γ(t - vx/c^2)

where:

• x, y, z are the spatial coordinates in the original frame S,
• t is the time in frame S,
• x', y', z' are the spatial coordinates in the moving frame S',
• t' is the time in frame S',
• v is the relative velocity between the two frames,
• c is the speed of light in a vacuum,
• γ (gamma) is the Lorentz factor, defined as γ = 1/√(1 - v^2/c^2).

These equations show that the measurements of space and time are not absolute but depend on the relative motion of the observer and the object being observed.

Consequences

The Lorentz transformations lead to several important consequences in the realm of physics, including:

• Time Dilation: Time intervals expand (dilate) in a moving frame of reference compared to a stationary frame. This means that a moving clock ticks slower than a stationary one, as observed from the stationary frame.
• Length Contraction: Objects contract in the direction of motion when moving at speeds close to the speed of light. An observer in a stationary frame would perceive a moving object to be shorter in the direction of its motion.
• Relativity of Simultaneity: Events that are simultaneous in one frame of reference may not be simultaneous in another frame moving relative to the first.

Impact on Physics

The Lorentz transformations have had a profound impact on our understanding of the universe. They are foundational to the theory of special relativity, which has been confirmed by numerous experiments and observations. Special relativity, in turn, has implications for a wide range of physical phenomena, from the behavior of particles in accelerators to the propagation of light in the universe, and it forms the basis for the theory of general relativity, Einstein's theory of gravitation.