Special Relativity: Time Dilation

Time Dilation

• Time appears to pass slower for an observer in motion relative to a stationary observer
• Time dilation is a consequence of special relativity's postulate that the speed of light is constant for all observers
• Mathematically, time dilation is represented by the equation: t = γ(t'), where t is time measured by the stationary observer, t' is time measured by the moving observer, and γ is the Lorentz factor
• Time dilation is a symmetric effect, meaning that each observer in relative motion will measure the other's clock as running slower

Length Contraction

• Objects appear shorter to an observer in motion relative to a stationary observer
• Length contraction is a consequence of special relativity's postulate that the speed of light is constant for all observers
• Mathematically, length contraction is represented by the equation: L = L0 / γ, where L is the length measured by the moving observer, L0 is the proper length (measured in the object's rest frame), and γ is the Lorentz factor
• Like time dilation, length contraction is a symmetric effect, meaning that each observer in relative motion will measure the other's objects as shorter

Relativity of Simultaneity

• Two events that are simultaneous for one observer may not be simultaneous for another observer in a different state of motion
• This is because the concept of "now" is relative, and depends on the observer's frame of reference
• The relativity of simultaneity is a direct consequence of time dilation and length contraction

Lorentz Transformation

• A mathematical formula that describes how space and time coordinates are transformed from one inertial frame of reference to another
• The Lorentz transformation is a linear transformation that preserves the spacetime interval between events
• The transformation is characterized by the Lorentz factor γ and the relative velocity v between the two frames of reference
• The Lorentz transformation is essential for making predictions in special relativity, as it allows us to calculate the spacetime coordinates of an event in one frame of reference from its coordinates in another frame

Invariant Mass

• A concept that describes the mass of an object as measured in its rest frame (i.e., when it is at rest relative to the observer)
• Invariant mass is a relativistic invariant, meaning that it has the same value for all observers, regardless of their relative motion
• Invariant mass is often denoted by the symbol m0 and is used to calculate the energy and momentum of an object in special relativity
• The concept of invariant mass is essential for making predictions in particle physics, as it allows us to calculate the properties of particles in high-energy collisions

Time Dilation

• Time appears to pass slower for an observer in motion relative to a stationary observer due to special relativity's postulate of constant light speed for all observers
• Mathematically represented by t = γ(t'), where t is time measured by the stationary observer, t' is time measured by the moving observer, and γ is the Lorentz factor
• A symmetric effect, where each observer in relative motion measures the other's clock as running slower

Length Contraction

• Objects appear shorter to an observer in motion relative to a stationary observer due to special relativity's postulate of constant light speed for all observers
• Mathematically represented by L = L0 / γ, where L is the length measured by the moving observer, L0 is the proper length, and γ is the Lorentz factor
• A symmetric effect, where each observer in relative motion measures the other's objects as shorter

Relativity of Simultaneity

• Two events simultaneous for one observer may not be simultaneous for another observer in a different state of motion due to relativity of "now"
• A direct consequence of time dilation and length contraction

Lorentz Transformation

• A mathematical formula describing the transformation of space and time coordinates from one inertial frame of reference to another
• A linear transformation preserving the spacetime interval between events
• Characterized by the Lorentz factor γ and the relative velocity v between two frames of reference
• Essential for making predictions in special relativity

Invariant Mass

• The mass of an object measured in its rest frame (when at rest relative to the observer)
• A relativistic invariant, with the same value for all observers regardless of relative motion
• Often denoted by m0, used to calculate energy and momentum of an object in special relativity
• Essential for making predictions in particle physics