Special relativity (3.12.3)
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The wave theory of light, first proposed by Christiaan Huygens, described light as a wave similar to sound or water waves. Since all known waves required a medium to travel through, scientists inferred that light must also propagate through a medium.
This hypothetical medium was named the luminiferous ether, an invisible substance thought to fill all space and provide an absolute frame of reference for all motion, including that of light.
According to Newtonian physics, the speed of light measured in a lab moving parallel to the light would vary depending on its direction relative to the ether wind, the hypothetical wind created by the Earth’s movement through the ether.
Consequently, light moving against the ether wind would travel slower than light moving perpendicular to it. This is similar to how a swimmer’s speed slows when moving against the current, or upstream.
The Michelson–Morley experiment, conducted in 1887, aimed to measure the difference in the speed of light when moving parallel and perpendicular to the Earth’s motion.
The expected change in the arrival time of each beam, caused by the ether wind, would create a phase shift in the interference pattern, which would be recorded by the observer.
This time difference would then be used to measure the absolute speed of the Earth relative to the stationary ether.
In 1887, Michelson and Morley used an interferometer to try to detect differences in light speed caused by Earth’s motion through the ether.
- In the experiment, monochromatic (single-wavelength) light is split into two beams using a beam splitter: One beam moves parallel to the motion, and the other moves perpendicular to it.
- The beams are then reflected at mirrors A and B and meet back at the beam splitter. A glass block is placed in the path of the transmitted beam to ensure both beams travel through the same amount of air and glass.
- The interference pattern, caused by the expected difference in arrival times of the beams, is recorded by an observer.
- The interferometer is then rotated by 90°, and the experiment is repeated.
Newtonian mechanics posits absolute space and time, with motion and speed measured relative to a fixed frame of reference. Therefore, a beam of light travelling parallel to Earth’s motion should take slightly longer to go to and from a mirror than a beam travelling perpendicular to the Earth’s motion. The Michelson–Morley experiment aimed to detect this difference caused by Earth’s motion through the ether.
In the experiment, rotating the interferometer should alter the travel times of the two beams, creating a phase difference and producing a small but measurable shift in the interference pattern. But, repeated null results throughout the year showed they had not detected a shift, providing strong evidence against the existence of the ether.
Three conclusions were drawn:
- The ether does not exist.
- It is impossible to detect absolute motion – we can only measure relative motion.
- The speed of light is invariant (constant) for all observers, regardless of their motion relative to the source.
A frame of reference is a coordinate system used to describe the position and motion of objects. It is essentially a point of view from which measurements are made, and different frames of reference can result in different interpretations of the same physical event.
In physics, all motion must be described relative to a chosen frame of reference.
For example, a traveller standing still inside a moving train appears stationary to observer A (in the same frame), but appears to be moving relative to observer B on the platform (in a different frame). The motion depends entirely on the observer’s frame of reference.
An inertial frame of reference is one in which Newton’s first law is obeyed – objects remain at rest or move with constant velocity unless acted on by an external force. In other words, it is a non-accelerating reference frame.
Frames of reference that are accelerating (changing velocity or direction) are called non-inertial frames.
An example of this is when a car accelerates quickly, and you feel pushed back into your seat. That’s not a real force, but an inertial, fictitious force that appears because you’re in an accelerating non-inertial frame. In reality, it’s the car that’s changing velocity relative to an inertial frame (the road), while you’re briefly resisting that change due to your inertia.
Question walkthrough
Identifying inertial and non-inertial frames
Analyses a ball’s motion relative to a vehicle in four scenarios (stationary, constant velocity, accelerating, and cornering) to identify which represent inertial versus non-inertial reference frames.
Don’t confuse constant velocity with constant acceleration. One is inertial, the other is not.
Einstein’s theory of special relativity applies only in inertial frames and is built on two fundamental postulates:
- The principle of relativity:
The laws of physics are the same in all inertial frames of reference, so all experiments yield the same results. No experiment performed in a uniform-motion frame can detect its absolute motion. In other words, it is impossible to tell whether your frame of reference is stationary or moving at a constant velocity. - Invariance of the speed of light:
The speed of light in a vacuum, is the same for all observers, regardless of the motion of the source or the observer.
The proper time is the time interval measured by a clock (or observer) at rest relative to the events being observed. In this situation, a stationary observer measuring two events with a clock records the shortest possible time interval between those events.
It is important to note that since absolute motion doesn’t exist, a ‘stationary’ observer simply means someone at rest relative to the frame in which the events occur.
Imagine a high-speed train travelling at a speed close to that of light. A passenger on the train may measure their journey to last 2 minutes according to the train’s clock. This is the proper time. However, an observer standing on the platform may see the same journey take longer, because the train’s clock is moving relative to them.
An observer moving relative to the events taking place measures a longer time interval than the proper time, since the moving clock appears to run slower. The difference is usually negligible at everyday speeds but becomes significant as the relative speed approaches the speed of light.
The (dilated) time interval between two events, measured by an observer moving relative to the frame in which the events occur, is given by:
Where:
- dilated time interval (s),
- proper time interval (s),
- relative speed between the observer and the reference frame and
- speed of light in a vacuum
It is useful to note that as the relative speed between the observer and the reference frame increases, the denominator in the time-dilation expression becomes smaller in a non-linear way. This increases the dilated time interval. The effect becomes increasingly pronounced as the speed approaches light speed, with the time dilation factor rising sharply toward infinity.
Question walkthrough
Finding proper time via time dilation
Uses t=d/s to find Earth’s measured travel time to an exoplanet, then t₀=t√(1−v²/c²) to find the shorter proper time experienced by the travelling astronaut.
Muons are unstable particles created in the Earth’s upper atmosphere that move toward the surface at speeds close to the speed of light. Their half-life at rest is very short so classically, almost all should decay before reaching the ground.
However, this is not the case; a considerably larger number of muons survive the journey than expected. These can be detected on the Earth’s surface, providing experimental evidence of time dilation.
The following experiment measures muon decay:
- Measure muon speed from rest in a non-inertial frame (expect ).
- Measure the muon count rate with a detector in the high atmosphere.
- Measure the muon count rate at ground level with a second detector.
- Compare count rates to calculate the muon survival number.
Experimental data indicate that far more muons survive than predicted by classical models based on their short half-life. Proper time is measured in the reference frame which is stationary relative to the events being measured. In this case, the event is the muons decaying, so proper time is measured in the muon’s frame of reference.
From the observer’s frame on Earth, time dilation appears to extend the muons’ lifetime, so more survive the journey. However, from the muon’s frame, it is at rest and measures the proper time for its decay. From the muon’s frame of reference, the journey actually takes less time than its half-life, which explains its survival.
Just as time passes at different rates for observers moving relative to each other, length is also relative; its value depends on the observer measuring it.
An object moving at a significant fraction of the speed of light will appear shortened in its direction of motion when measured by a stationary observer. This effect, known as length contraction, is another direct consequence of special relativity.
It is important to note that the observer in motion in the diagram above is stationary in their own reference frame. Therefore, they would still observe length contraction of all moving objects in their direction of motion.
The proper length is the length of an object measured in the frame of reference where the object is at rest. A stationary observer in this frame of reference will always record the maximum length.
To an observer who observes the object as moving, the measured length is shorter.
It is important to note that since absolute rest doesn’t exist, at rest simply means the object is stationary relative to the observer making the measurement.
A passenger on a high-speed train, travelling close to the speed of light, may measure the length of the train to be using a long tape measure at rest with the train. This is the proper length of the train. However, to an observer standing on the platform, the moving train appears shorter along the direction of motion. This is because the observer on the platform is moving relative to the train.
An observer moving relative to an object will always measure a contracted (shorter) length compared to the proper length. Like time dilation, the effect is negligible at everyday speeds, but becomes significant as the relative speed approaches the speed of light.
The (contracted) length of an object, measured by an observer moving relative to the frame in which the object is stationary, is given by:
Where:
- contracted length (m),
- proper length (m),
- relative speed between the observers and
- speed of light in a vacuum
It is useful to note that as the relative speed between the observer and the reference frame increases, the term within the square root of the equation becomes smaller in a non-linear way. This causes the contracted length to decrease in size. As the relative speed approaches the light speed, this effect becomes more significant, with the contracted length asymptotically nearing zero.
Question walkthrough
Finding proper length via length contraction
Rearranges the length contraction equation l=l₀√(1−v²/c²) to find a spaceship’s proper length (measured by an astronaut aboard) from its Earth-observed contracted length and speed.
Einstein established the equivalence of mass and energy, demonstrating that they are not distinct entities but rather different manifestations of the same phenomenon. This relationship is famously captured by the equation:
Where:
- mass of the object (kg), and
- speed of light in a vacuum
This formula implies that mass and energy are interchangeable, meaning that mass can be converted into energy and vice versa. It also implies that supplying energy to an object effectively increases its mass, although the change is normally far too small to detect in everyday situations.
Question walkthrough
Finding photon energy from annihilation
Uses E=mc² to find the total energy released in an electron–positron annihilation, then halves it to find the energy of each of the two emitted photons.
The rest mass is the mass of an object measured in the frame of reference where the object is at rest. A stationary observer in this frame of reference will always record the minimum value of the mass of an object.
Rest mass is an invariant and fundamental property of matter, independent of the object’s speed or the observer’s reference frame.
The relativistic mass of an object is the apparent mass measured by an observer in an inertial reference frame moving at a speed relative to the object. Its value appears to increase as the relative velocity between the object and the observer increases. This means that an observer moving relative to an object will always measure the object’s relativistic mass as greater than its rest mass.
Relativistic mass is given by the formula:
Where:
- relativistic mass of the object (kg),
- rest mass of the object (kg),
- relative speed between the observers and
- speed of light in a vacuum
It is useful to note that as the relative speed between the observer and the reference frame increases, the denominator in the relativistic mass equation decreases in a non-linear manner. This causes the relativistic mass to increase. The effect becomes increasingly pronounced as the speed approaches light speed, with the relativistic mass rising sharply toward infinity.
As an object’s speed increases, its relativistic mass also increases. On a plot of the relativistic mass against speed, the curve rises very slowly at first. However, as the object approaches the speed of light, its relativistic mass increases significantly and tends toward infinity.
The rest mass of an object remains constant, but to an external observer, its relativistic mass increases without limit as the object approaches relativistic speeds.
The relativistic kinetic energy of an object increases at a much faster rate than predicted by the classical equation.
On a plot of kinetic energy against speed, the curve rises sharply as the object approaches the speed of light. This indicates that more and more energy is required for increasingly smaller increases in speed.
It is important to note that an object can approach the speed of light but never reach or exceed it, regardless of the amount of energy supplied to the object.
Question walkthrough
Explaining why light speed is unreachable
Uses the relativistic mass equation m=m₀/√(1−v²/c²) to explain why an object with mass can never be accelerated to the speed of light.
Relativistic energy is the total energy of a moving object, which includes both its intrinsic rest energy (due to its mass) and any additional kinetic energy (due to its motion). It can be calculated using the formula:
Where:
- rest mass of the object (kg),
- relative speed between the observers and
- speed of light in a vacuum
The value of an object’s relativistic energy increases significantly as the object approaches the speed of light.
Question walkthrough
Finding relativistic energy of an electron
Uses E=m₀c²/√(1−v²/c²) to find the total relativistic energy of a fast-moving electron, then converts the result from joules to electron-volts.
In the 1960s, William Bertozzi conducted an experiment to directly test Einstein’s theory of relativity.
- He accelerated electrons to speeds close to the speed of light using a linear particle accelerator, where their velocities were determined by measuring the time it took them to travel a set distance.
- The electrons were then smashed into an aluminium disc, where their kinetic energy was converted into heat.
- By measuring the disc’s temperature rise, Bertozzi determined the electrons’ kinetic energy immediately before impact.
The results of Bertozzi’s experiment were plotted on a graph of the squared speeds of the electrons against their kinetic energy, represented by the blue crosses.
The data showed that the kinetic energy of the electrons initially increased rapidly with speed, but eventually levelled off as the electrons approached the speed of light.
The experimental data closely followed the relativistic prediction, represented by the green line.
A dotted line at represents an asymptote, illustrating that no matter how much energy was provided, the speed of the electrons could never reach or exceed the speed of light.
The red line represents the Newtonian prediction, based on the classical formula This predicted that the speed of the electrons would continue to increase linearly and without limit.
It is clear that the Newtonian prediction and experimental data differ significantly.
Three important conclusions were drawn from Bertozzi’s experiment:
No object can exceed the speed of light:
- As electrons’ kinetic energy increased, their velocity approached the speed of light but never reached or exceeded it.
Relativistic mass increases with velocity:
- At relativistic speeds, additional energy input produced only a negligible increase in the electrons’ velocity. However, their kinetic energy continued to rise significantly, providing direct evidence for relativistic mass, as further energy input increases mass rather than speed at high velocities.
Special relativity is more accurate than Newtonian mechanics:
- Bertozzi’s results closely matched the predictions of special relativity, contradicting the predictions of Newtonian mechanics at relativistic speeds. Newtonian mechanics remain an excellent approximation in everyday situations. Special relativity corrects these limitations, providing a more comprehensive and accurate theory of motion.