Relativistic kinematics

Special Theory of Relativity

In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time.

Postulates Of Special relativity

1. First postulate (principle of relativity)

   The laws of physics take the same form in all inertial frames of reference.

2. Second postulate (invariance of c) As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

simultaneity Is a Relative Concept not an Absolute - Prove IT

Effect Of Special Relativity to Be significant

Lorentz Transformation & consequences

Galilean Transformation vs. Lorentz Transformation: Similarities and Differences

Galilean and Lorentz transformations are both methods used to describe the motion of objects, but they apply under different circumstances and have significant differences in their implications. Here’s a breakdown to understand them better:

Galilean Transformation:

• Applicability: Valid for objects moving at speeds much slower than the speed of light (c). It is a classical mechanics approach.
• Assumptions:
  • Space and time are absolute and independent.
  • Velocity is simply the sum of relative velocities.
• Equations:
  • x’ = x - vt
  • y’ = y
  • z’ = z
  • t’ = t
  • Where:
    • x, y, z are original coordinates
    • x’, y’, z’ are new coordinates after transformation
    • v is relative velocity
    • t is original time
    • t’ is new time
• Consequences:
  • Time is universal and independent of observer’s motion.
  • Lengths and distances are not affected by relative motion.
  • Velocity addition is simply linear.

Lorentz Transformation:

• Applicability: Valid for objects moving at any speed, including speeds close to or exceeding the speed of light (c). It is a relativistic mechanics approach.
• Assumptions:
  • Space and time are relative and interconnected.
  • The speed of light is constant in all reference frames.
• Equations:
  • x’ = γ (x - vt)
  • y’ = y
  • z’ = z
  • t’ = γ (t - vx/c^2)
  • Where:
    • γ = 1 / sqrt(1 - v^2/c^2) (Lorentz factor)
    • Other variables are the same as in Galilean transformation.
• Consequences:
  • Time dilation: Time slows down for moving objects compared to a stationary observer.
  • Length contraction: Moving objects appear shorter in the direction of motion.
  • Velocity addition is non-linear and involves the Lorentz factor.

Key Differences:

• Range of validity: Galilean for slow speeds, Lorentz for any speed.
• Space-time: Galilean assumes absolute space and time, Lorentz assumes relative and interconnected.
• Consequences: Galilean has no time dilation or length contraction, Lorentz does.
• Equations: Lorentz transformation involves the Lorentz factor and is more complex.

Choosing the Right Transformation:

• For everyday objects and speeds much slower than the speed of light, the Galilean transformation is a good approximation.
• For high-speed objects, near or exceeding the speed of light, the Lorentz transformation is essential for accurate predictions.

Understanding these differences is crucial in various fields like particle physics, astrophysics, and high-speed engineering, where relativistic effects become significant.