A.5. Galilean and Special Relativity (HL Only)

An inertial frame is one in which Newton's laws hold true; it is either at rest or moves with constant velocity.

An inertial frame of reference is a perspective or viewpoint from which you can observe motion. In such a frame, the laws of physics, particularly Newton's laws of motion, are valid. This means that if you are in a car moving at a constant speed on a straight road, you are in an inertial frame. Newton's laws apply to you because there are no external forces acting on you. Generally, inertial frames are at rest or have uniform motion without acceleration.

Imagine you are sitting in a train that is moving steadily along the tracks. You can toss a ball in the air, and it falls straight back into your hand. In this scenario, the train is your inertial frame. Now, contrast this with being in a train that is accelerating or turning; the ball would not come back to you in a straight line due to the effects of acceleration, and that scenario does not represent an inertial frame.

Classical transformations between inertial frames moving at relative velocity vvv:

● x′=x−vtx' = x - vtx′=x−vt ● y′=yy' = yy′=y ● z′=zz' = zz′=z ● t′=tt' = tt′=t

These assume absolute time and are valid at speeds much less than the speed of light.

Galilean transformations are mathematical equations used to relate coordinates (and time) of events as observed in two different inertial frames moving with a constant relative velocity. The equations show how the position (x, y, z) and time (t) in one frame can be converted into another frame's coordinates. For example, if one frame is moving at a constant speed 'v,' the position in the moving frame (x') is found by subtracting the product of velocity and time from the stationary position (x). These transformations are mainly used when velocities are much smaller than the speed of light, indicating that time is considered absolute.

Think of two cars driving on a parallel road. If one car is moving at a uniform speed (let's say a speed of 60 km/h) and the other is stationary, the driver of the moving car can use Galilean transformations to determine the relative positions of both cars over time. The transformations help them understand their motion relative to each other, much like how physics applies to each car but does not include the effects of extremes like light speed.

● First Postulate: The laws of physics are the same in all inertial frames. ● Second Postulate: The speed of light in a vacuum is constant for all observers, regardless of their relative motion.

The postulates of special relativity establish fundamental principles for understanding how the laws of physics apply universally. The first postulate asserts that if you perform an experiment in any inertial frame, the results will be the same as in any other inertial frame. The second postulate states that no matter how fast you are moving or what direction you're going, the speed of light remains constant at approximately 299,792 kilometers per second. This revelation overturns previous assumptions about motion and speeds and sets the groundwork for Einstein's theory.

Imagine two observers: one stationary on the ground and another flying in a spaceship. If a beam of light passes them both, both observers will measure the speed of that light beam to be the same, even though the spaceship is moving quickly in a different direction. This constancy of light speed applies to anything we measure regarding motion, reshaping how we view interactions in our universe.

Moving clocks run slower compared to stationary ones.

Δt=Δt01−v2c2 Where: ● Δt: Time interval measured by a stationary observer ● Δt0: Proper time interval (measured in the moving frame) ● v: Relative velocity ● c: Speed of light

Time dilation is a result of special relativity that states that time intervals appear longer for objects moving at high speeds compared to those who are stationary. The equation shows that the time interval (Δt) observed from a stationary frame is affected by the speed of the moving clock (v) relative to the speed of light (c). If an object is moving very fast (close to the speed of light), the time measured by that moving object's clock (Δt0) will be shorter when observed from a stationary point of view.

Consider a scenario where astronauts travel in a spaceship at a significant fraction of the speed of light. If they set their clocks for a mission lasting 5 years according to their onboard time, upon returning to Earth, they may find that many more years have passed on Earth. This emphasizes the 'slowing down' of time for the astronauts relative to those who remained on Earth—their clocks ran slower due to their high-speed travel.

Objects moving at high speeds appear contracted along the direction of motion.

L=L01−v2c2 Where: ● L: Length measured by a stationary observer ● L0: Proper length (measured in the object's rest frame)

Length contraction is another fascinating consequence of special relativity, indicating that an object's length will appear shorter when it is moving fast relative to an observer. The equation shows that the observed length (L) becomes less than the proper length (L0) as the object's speed (v) approaches the speed of light (c). This reduction only happens along the direction of motion and is imperceptible at low speeds.

Visualize a train speeding down the tracks at a substantial fraction of light speed. An observer standing still at the station will see the length of the train shorter than it is at rest due to its rapid motion—a phenomenon that doesn't affect the passengers aboard. Thus, the contracted length is entirely a byproduct of their relative motion, showcasing how perceptions of space differ in relativistic terms.

At speeds approaching the speed of light, momentum is given by:

p=mv1−v2c2 This accounts for the increase in mass with velocity.

Relativistic momentum extends the classical notion of momentum (mass times velocity) by incorporating the effects of relativity at high speeds. As an object's velocity approaches the speed of light, its momentum (p) increases not just because its velocity increases but because its effective mass increases as well. The equation highlights that momentum does not behave linearly at relativistic speeds, necessitating the addition of a factor derived from the velocity's square over the speed of light's square.

Think of a sports car that accelerates quickly; it gains momentum and speed. Now imagine a spaceship approaching light speed—its momentum becomes significantly larger than would be calculated using traditional equations because of this relativistic factor. This shift in momentum behavior has crucial implications for particle physics, where fast-moving particles exhibit energy and behavior that defy classic interpretations.

--