LENGTH CONTRACTION

• Length contraction is a fundamental concept in special relativity that describes the apparent contraction of an object’s length to an observer in motion relative to the object. Here are the key aspects of length contraction:
• – Apparent contraction: The length of an object appears shorter to an observer in motion relative to the object.
• – Length contraction factor: The apparent length contraction is given by the factor Where;
• [math]l = l_0 \sqrt{1 – \frac{v^2}{c^2}}[/math]
•  v is the velocity of the at which the object is travelling.
• It is important to note that even though the length of an object moving relative to an external observer will appear shorter, its width will remain constant as only length is affected.
• – Proper length: The proper length of an object is its length measured by an observer at rest with respect to the object.
• – Contracted length: The contracted length is the apparent length measured by an observer in motion.
• Muon decay is a great example to illustrate length contraction. Here’s how it works:
• – Muons are created in flight, traveling at high speeds (approximately 98% of the speed of light).
• – Muons decay into electrons, muon neutrinos, and antimuon neutrinos.
• – The decay length of muons is measured, which is the distance they travel before decaying.
• – Due to length contraction, the decay length appears shorter to an observer in motion (the muon) than to an observer at rest (the laboratory frame).
• Let’s use the length contraction equation:
• [math]l = l_0 \sqrt{1 – \frac{v^2}{c^2}}[/math]
• Where:
• [math]l[/math]= contracted decay length (measured by the muon)
• [math]l_0[/math]= proper decay length (measured in the laboratory frame)
• [math]\sqrt{1 – \frac{v^2}{c^2}}[/math]= length contraction factor
• Plugging in the values, we get:
• [math]l \approx 0.6 l_0[/math]
• This means that the muon’s decay length appears approximately 60% of its proper length due to length contraction.
• Since the muon is traveling at high speed, its decay length is contracted in the direction of motion. This results in a shorter decay length measured by the muon compared to the laboratory frame.
• – Particle accelerator: Particles accelerated to high speeds experience length contraction. For instance, a proton traveling at 90% of the speed of light will appear approximately 66% of its proper length to an observer.
• – High-speed train: Imagine two observers, Alice and Bob. Alice is on a train traveling at 80% of the speed of light relative to Bob, who is standing on the platform. They both measure the length of the train. Due to length contraction, Alice will measure a shorter length than Bob.
• – Astronomical objects: Distant galaxies and stars appear smaller due to length contraction. For example, a galaxy 100,000 light-years away will appear approximately 60% of its proper length to an observer moving at 80% of the speed of light relative to the galaxy.
• – Relativistic jets: Jets of particles ejected from black holes or neutron stars can travel at significant fractions of the speed of light. Length contraction causes these jets to appear shorter to observers in motion relative to the jets.

Mass and energy

• ⇒ Equivalence of mass and energy:

• The equivalence of mass and energy is a fundamental concept in physics, and the famous equation [math]E = mc^2[/math] represents this idea.
• [math]E^2 = (pc)^2 + (mc^2)^2[/math]
• Where:
• – E is the total energy of the object
• – p is the momentum of the object
• – m is the rest mass of the object
• – c is the speed of light
• This equation shows that energy (E) is equal to the sum of the kinetic energy [math](pc)^2[/math] and the rest energy [math](mc^2)^2[/math].
• However, if we consider an object at rest (p=0), the equation simplifies to:
• [math]E = mc^2[/math]
• This is the famous equation that shows that mass (m) is equivalent to energy (E).
• [math]E = \frac{mc^2}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
• Is actually the energy-momentum relation for an object in motion, where v is the velocity of the object. This equation shows that the energy of an object in motion is greater than its rest energy ([math](mc^2)^2[/math] ) due to the additional kinetic energy.
• So, to summarize:
• [math]E = mc^2[/math]
• – Represents the equivalence of mass and energy for an object at rest
• [math]E^2 = (pc)^2 + (mc^2)^2[/math]
• – Represents the energy-momentum relation for an object in motion.

• ⇒ Graphs of variation of mass and kinetic energy with speed:

• Mass vs. velocity:
• – The graph shows the variation of mass with velocity.
• – At low velocity (v < 0.5c), the mass remains constant ([math]m = m_0[/math]).
• – As velocity increases (0.5c < v < 0.9c), the mass starts to increase gradually.
• – At high velocity (v > 0.9c), the mass increases rapidly, approaching infinity as v approaches c.
• This graph illustrates the concept of relativistic mass, which increases as an object approaches the velocity of light.
• 
• Figure 1 Mass and velocity graph
• Kinetic Energy vs. Speed:
• – The graph shows the variation of kinetic energy with speed.
• – At low speeds (v < 0.5c), the kinetic energy increases quadratically with speed ([math]KE \propto v^2[/math] ).
• – As speed increases ( 0.5c < v < 0.9c), the kinetic energy increases more rapidly, but still remains below the relativistic energy ([math]KE < mc^2[/math] ).
• – At high speeds (v > 0.9c), the kinetic energy approaches the relativistic energy (KE → [math]mc^2[/math]), and eventually exceeds it.
• 
• Figure 2 Kinetic energy Vs speed graph
• This graph demonstrates how kinetic energy increases rapidly as an object approaches the speed of light, eventually approaching the energy equivalent of its rest mass ([math]mc^2[/math] ).
• The total energy of a relativistic object can be calculated using the following formula:
• [math]E = \frac{mc^2}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
• These graphs illustrate the strange and counterintuitive consequences of special relativity, where mass and energy become intertwined, and the rules of classical physics no longer apply.
• As [math]\text{total energy } (E_T) = \text{kinetic energy } (E_k) + \text{rest energy } (E_0)[/math] , you can use the formula below to find the kinetic energy of an object moving at relativistic speeds;
• [math]\text{kinetic energy } (E_k) = \text{total energy } (E_T) – \text{rest energy } (E_0) \\
  \text{total energy } (E_k) = \frac{m_0 c^2}{\sqrt{1 – \frac{v^2}{c^2}}} – m_0 c^2[/math]
• Where v is the velocity of the at which the object is travelling and [math][/math] is rest mass.
• Bertozzi’s experiment as direct evidence for the variation of kinetic energy with speed:
• In 1964, physicist William Bertozzi performed an experiment to test the relativistic energy-momentum relation. He accelerated electrons to high speeds and measured their energy and momentum. The experiment confirmed the relativistic energy-momentum relation:
• [math]E^2 = (pc)^2 + (mc^2)^2[/math]
• Where E is the total energy, p is the momentum, m is the rest mass, and c is the speed of light.
• – Bertozzi accelerated electrons to speeds up to 90% of the speed of light using a particle accelerator.
• – He measured the energy of the electrons using a spectrometer.
• – He measured the momentum of the electrons by deflecting them in a magnetic field.
• – He compared the measured energy and momentum values to the predicted values from the relativistic energy-momentum relation.
• 
• Figure 3
• The experiment confirmed that the relativistic energy-momentum relation holds true, even at high speeds. This experiment provided strong evidence for the validity of special relativity and has since been repeated and verified by numerous other experiments.
• Graphical representation of the variation of kinetic energy with speed, illustrating the relativistic nature of kinetic energy:
• Kinetic Energy (KE) vs. Speed (v)
• – Low speeds (v < 0.5c): KE increases quadratically with speed ([math]KE \propto v^2[/math] )
• – Relativistic speeds (0.5c < v < 0.9c): KE increases more rapidly, deviating from classical behavior
• High speeds (v > 0.9c): KE approaches the energy equivalent of the rest mass (KE → [math]mc^2[/math] )
• 
• Figure 4
• Bertozzi’s experiment confirms this relativistic behavior, providing direct evidence for the variation of kinetic energy with speed.
• Bertozzi’s experiment marked an important milestone in the history of physics, demonstrating the power of experimentation in testing fundamental theories like special relativity.