Galilean & Special Relativity (DP IB Physics)

The Galilean transform equations are used to convert velocity, position, and time between reference frames.

The position and time in reference frame are denoted as , , and .

In reference frame , these are , , and .

The Galilean transformation equation from to is:

Where:

• = coordinate in the moving reference frame

• = coordinate in the stationary reference frame

• = velocity of the moving reference frame, , with respect to the stationary reference frame,

• = time at which the event occurred

The Galilean transformation equation from to is:

The coordinates of are the same in both reference frames because the relative motion is only ever considered in the direction.

The Galilean transformation equation from to is:

The coordinates of are the same in both reference frames because the relative motion is only ever considered in the direction.

The Galilean transformation equation from to is:

The coordinates of are the same in both reference frames because the origin of both reference frames coincide at . Both observers agree on the time of the event, but disagree on the position that event occurs at.

The Galilean transformation equation from to is:

Where:

• = coordinate in the stationary reference frame

• = coordinate in the moving reference frame

• = velocity of the moving reference frame, , with respect to the stationary reference frame,

• = time at which the event occurred

The Galilean transformation equation from to is:

Where:

• = velocity of an object, as measured from within the moving reference frame,

• = velocity of an object, as measured from the stationary reference frame,

• = velocity of the moving reference frame, , with respect to the stationary reference frame,

The Galilean transformation equation from to is:

Where:

• = velocity of an object, as measured from the stationary reference frame,

• = velocity of an object, as measured from within the moving reference frame,

• = velocity of the moving reference frame, , with respect to the stationary reference frame,

Einstein's two postulates of special relativity are:

• the laws of physics are the same in all inertial reference frames

• the speed of light in a vacuum is the same in all inertial reference frames

Special relativity applies when objects are travelling close to the speed of light when time and space are also relative to the frame of reference.

True.

Two different observers will always measure the speed of light to be the same value in their own reference frames.

True.

Velocity addition, or the Galilean transformation for the velocity of an object moving within the moving reference frame, only works at speeds much lower than the speed of light.

True.

Galilean transformations predict that objects can travel faster than the speed of light, but this cannot be the case, so for objects travelling at speeds close to the speed of light, Lorentz transformations must be used to correct this.

The Lorentz transformations are corrections to the Galilean transformations used when objects are travelling close to the speed of light.

The equation for the Lorentz factor is

Where:

• = Lorentz factor (no units)

• = velocity of moving object, measured in metres per second (m s^–1)

• = speed of light in a vacuum, measured in metres per second (m s^–1)

False.

The Lorentz factor for any moving object will always be greater than 1.

This is because the speed of any moving object will always be less than the speed of light.

The Lorentz transformation equation for to is:

Where:

• = coordinate as measured from the moving reference frame,

• = Lorentz factor (no units)

• = coordinate as measured from the stationary reference frame,

• = speed of moving reference frame, , relative to the stationary reference frame,

• = time of the event as measured from the stationary reference frame,

The Lorentz transformation equation for to is:

Where:

• = coordinate as measured from the stationary reference frame,

• = Lorentz factor (no units)

• = coordinate as measured from the moving reference frame,

• = speed of moving reference frame, , relative to the stationary reference frame,

• = time of the event as measured from the moving reference frame,

The Lorentz transformation equation for to is:

Where:

• = time of the event as measured from the stationary reference frame,

• = Lorentz factor (no units)

• = time of the event as measured from the moving reference frame,

• = speed of the moving reference frame, , relative to the stationary reference frame,

• = coordinate as measured from the moving reference frame,

• = speed of light, constant in both reference frames

The Lorentz transformation equation for to is:

Where:

• = time of the event as measured from the moving reference frame,

• = time of the event as measured from the stationary reference frame,

• = Lorentz factor (no units)

• = time of the event as measured from the stationary reference frame,

• = speed of the moving reference frame, , relative to the stationary reference frame,

• = coordinate as measured from the stationary reference frame,

• = speed of light, constant in both reference frames

The Lorentz transformation equation from to is:

The coordinates of are the same in both reference frames because the relative motion is only ever considered in the direction.

The Lorentz transformation equation from to is:

The coordinates of are the same in both reference frames because the relative motion is only ever considered in the direction.

The Lorentz velocity addition equation for to is:

Where:

• = velocity of an object, as measured from the moving reference frame,

• = velocity of an object, as measured from the stationary reference frame,

• = velocity of moving reference frame, , relative to the stationary reference frame,

• = speed of light, constant in both reference frames

The Lorentz velocity addition equation for to is:

Where:

• = velocity of an object, as measured from the stationary reference frame,

• = velocity of an object, as measured from the moving reference frame,

• = velocity of moving reference frame, , relative to the stationary reference frame,

• = speed of light, constant in both reference frames

The Lorentz velocity addition transformations should be used when

• an object is moving

• both the object and the moving reference frame are travelling close to the speed of light

A rocket ship travelling close to the speed of light launches a probe, which also travels close to the speed of light. A stationary observer on a nearby planet witnesses the launch of the probe.

• = the velocity of the probe as measured by the stationary observer on the nearby planet

• = the velocity of the probe as measured by an observer in the moving rocket ship

• = the velocity of the moving space rocket relative to the stationary observer on the nearby planet

• = the stationary reference frame of the observer on the nearby planet

• = the moving reference frame of the observer on the moving rocket ship

The speed of the probe as measured by a stationary observer on a nearby planet is

A rocket ship travelling at launches a probe, which travels at as measured by the pilot of the rocket ship, . A stationary observer, , on a nearby planet witnesses the launch of the probe.

Invariant quantities are quantities that are the same in all inertial reference frames.

The invariant quantities are:

• proper time

• proper length

• space-time interval

• speed of light in a vacuum

The space-time interval is the change in space-time between two events as represented on a four-dimensional coordinate system (, , , ) in special relativity. The space-time interval is an invariant quantity that is the same in all inertial reference frames.

The equation for the space-time interval is

Where:

• = space-time interval, measured in metres (m)

• = speed of light in a vacuum, measured in metres per second (m s^-1)

• = temporal separation, measured in seconds (s)

• = spatial separation, measured in metres (m)

Proper length is the length of an object measured in a reference frame where the object is at rest relative to the observer.

Proper time is the time interval between two events measured in the reference frame where the events occur in the same place. Proper time is measured by an observer within their own rest frame.

True.

Motion can be described as spanning both time and space using a 4D coordinate system (, , , ); this is the space-time interval.

True.

The units of the space-time interval are metres.

Time dilation is the apparent slowing of time for an object moving at close to the speed of light as measured by a stationary observer.

The equation for time dilation is

Where:

• = dilated time interval, as measured by an observer moving relative to the clock being used to measure proper time

• = Lorentz factor

• = proper time interval, as measured by the clock in its own rest frame

True.

For a spaceship travelling at close to the speed of light, time slows down on the spaceship from the reference frame of a stationary observer on Earth.

True.

For a spaceship travelling at close to the speed of light, time slows down on Earth from the reference frame of the spaceship because the spaceship is stationary in its own reference frame and Earth is moving at close to the speed of light relative to them.

The observer measuring proper time in this situation is the passenger on the train. The clock being used to measure the time in both reference frames is the clock on the train.

Therefore, proper time is the time measured by the clock on the train in its own rest frame.

The observer who measures proper time in this situation is the person on the platform. The clock being used to measure the time in both reference frames is the clock on the platform.

Therefore, proper time is the time being measured by the clock on the platform in its own rest frame.

Length contraction is the apparent shortening of the length of an object moving at close to the speed of light as measured by a stationary observer.

The equation for length contraction is

Where:

• = contracted length, as measured by an observer moving relative to the object being measured

• = Lorentz factor

• = proper length, as measured in the object's own rest frame

True.

For a spaceship travelling at close to the speed of light, the length of an object on the spaceship appears shorter from the reference frame of a stationary observer on Earth.

True.

For a spaceship travelling at close to the speed of light, the length of an object on Earth appears shorter from the reference frame of the spaceship.

This is because the spaceship is stationary in its own reference frame and Earth is moving at close to the speed of light relative to them.

True.

Length measurements for length dilation calculations are assumed to be taken with a ruler within each observer's reference frame.

This is because the stationary observer would also see the moving ruler contract as well as the object it is measuring.

True.

Length measurements in length contraction calculations are assumed to be taken at the same time in each reference frame.

The observer measuring proper length in this situation is the passenger on the train.

Moving objects contract along their direction of motion. Therefore, proper length is the length measured from the measured object's rest frame. The contracted length is measured by the observer at rest to the moving object.

The observer measuring proper length in this situation is the person on the platform.

Moving objects contract along their direction of motion. Therefore, proper length is the length measured from the measured object's rest frame. The contracted length is measured by the observer at rest to the moving object; for the person on the train, the platform is moving relative to them.

Relativity of simultaneity refers to the fact that two spatially simultaneous events do not necessarily occur at the same time, depending on the observer's reference frame.

True.

According to special relativity, two events that occur simultaneously in one reference frame may not occur simultaneously in another reference frame. This is also referred to as the relativity of simultaneity.

True.

In Galilean relativity, simultaneity is absolute. If two events occur at the same time in one reference frame, then they also occur at the same time in other reference frames.

True.

On a spacetime diagram, the three dimensions of space are collapsed into one dimension.

True.

On a spacetime diagram, the dimensions of time and space both have units of metres.

True.

On a spacetime diagram, the time axis is labelled as .

A line on a spacetime digram is known as a world line.

True.

On a spacetime diagram, the steeper the gradient, the slower the motion of the object through spacetime.

The gradient of a world line represents the velocity of the object.

Object Y is at rest in this spacetime diagram. It moves through time () with no change in its position ().

The fastest object is that with the shallowest gradient, so Object Z has the fastest speed.

The fastest speed an object can have is the speed of light. when the worldline is at a 45° angle from the axes. Any gradient shallower than 1 is not possible.

The relativistic phenomena that can be shown on a spacetime diagram are:

• multiple reference frames

• simultaneity

• time dilation

• length contraction

The muon lifetime experiment provides evidence for time dilation and length contraction.

In the muon lifetime experiment, the muon measures proper time, and the observer on Earth measures dilated time.

In the muon lifetime experiment, the observer on Earth measures proper length, and the muon measures contracted length.

Proper length is the length measured from the measured object's rest frame. The measured object is the distance between the atmosphere where the muons are produced and the detectors. The rest frame is therefore the stationary observer on Earth.

True.

Muons are produced in the upper atmosphere as a result of pion decays produced by cosmic rays.

True.

Muons travel at an average speed of .

True.

Muons have an average lifetime of 2.2μs.