1.3 Galilean relativity and its limitations

Galilean relativity describes how motion looks from different viewpoints. It works well for everyday speeds but falls short when things move super fast. This concept is crucial for understanding why we needed a new way to think about motion.

Einstein's special relativity came to the rescue, fixing the problems with Galilean relativity. It changed our view of space and time, showing that they're not as fixed as we thought. This shift was a big deal in physics.

Galilean Relativity

Coordinate Transformations and Relative Motion

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• Galilean transformations describe the relationship between coordinates in two inertial frames moving at constant velocity relative to each other
  • Consists of a translation in space and time
  • Allows for the conversion of coordinates from one inertial frame to another
• Inertial frames are reference frames in which Newton's laws of motion hold
  • An object at rest or moving at constant velocity remains at rest or moving at constant velocity unless acted upon by an external force
  • Experiments performed in one inertial frame will yield the same results as in any other inertial frame (principle of relativity)
• Relative motion refers to the motion of an object as observed from different inertial frames
  • The velocity of an object depends on the choice of the reference frame
  • An object at rest in one frame may appear to be moving in another frame

Velocity Addition and Invariance

• Velocity addition is a consequence of Galilean transformations
  • The velocity of an object in one inertial frame is the sum of its velocity in another frame and the relative velocity between the two frames
  • Formula for velocity addition: $v' = v + u$, where $v'$ is the velocity in the new frame, $v$ is the velocity in the original frame, and $u$ is the relative velocity between frames
• Galilean relativity implies the invariance of mechanical laws under Galilean transformations
  • The laws of mechanics have the same form in all inertial frames
  • Examples include Newton's laws of motion and the conservation of momentum

Invariance and Absolute Time

Acceleration and Its Invariance

• Acceleration is the rate of change of velocity over time
  • Mathematically, acceleration is the second derivative of position with respect to time: $a = d^2x/dt^2$
• Acceleration is invariant under Galilean transformations
  • The acceleration of an object is the same in all inertial frames
  • This invariance is a consequence of the absolute nature of time in Galilean relativity

Absolute Time and Simultaneity

• Galilean relativity assumes the existence of absolute time
  • Time is considered to flow at the same rate for all observers, regardless of their motion
  • The time interval between two events is the same in all inertial frames
• Absolute time implies the notion of absolute simultaneity
  • If two events are simultaneous in one inertial frame, they are simultaneous in all inertial frames
  • The order of events is preserved across all inertial frames

Limitations of Galilean Relativity

Inconsistencies at High Velocities

• Galilean relativity breaks down when objects move at speeds comparable to the speed of light
  • The invariance of the speed of light, as observed in experiments such as the Michelson-Morley experiment, cannot be explained by Galilean transformations
  • The addition of velocities in Galilean relativity would imply that the speed of light should vary depending on the observer's motion, which contradicts experimental evidence
• The limitations of Galilean relativity led to the development of special relativity by Albert Einstein
  • Special relativity postulates the invariance of the speed of light and the relativity of simultaneity
  • It introduces Lorentz transformations, which replace Galilean transformations, to describe the relationship between coordinates in different inertial frames moving at high velocities relative to each other