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28.2 Simultaneity And Time Dilation

3 min read•june 18, 2024

and relative motion are mind-bending concepts in physics. They challenge our everyday notions of time and space, revealing that events happening at the same moment for one observer might not be simultaneous for another.

and are wild consequences of these ideas. As objects move faster, time slows down for them and they appear shorter to outside observers. These effects become significant at speeds approaching light speed.

Simultaneity and Relative Motion

Simultaneity in reference frames

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  • Simultaneity describes events happening at the same instant in a given (observer's perspective)
  • of simultaneity means events simultaneous in one frame may not be in another due to different motion
  • Constant speed of light in all frames causes relativity of simultaneity (vacuum)
  • Observers in relative motion perceive simultaneity differently (moving train vs stationary platform)
  • method used to establish simultaneity in a given frame

Time dilation and relative motion

  • causes time to pass slower for objects moving at high speed relative to a stationary observer (cosmic ray muons)
  • Consequence of constant light speed and relativity of simultaneity (universal speed limit)
  • Time dilation increases as object's speed approaches light speed (relativistic effects)
  • (γ\gamma) relates time dilation to relative velocity between moving object and observer (mathematical description)
  • between events experiences time dilation in different reference frames

Time Dilation Calculations and Paradoxes

Calculations with Lorentz factor

  • Lorentz factor formula: γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} where vv is relative velocity and cc is light speed
  • Calculate (tt) by multiplying (t0t_0) by Lorentz factor: t=γt0t = \gamma t_0
  • Greater relative velocity leads to larger Lorentz factor and more pronounced time dilation (GPS satellites)

Proper vs dilated time

  • (t0t_0) measured by clock in object's own rest frame (astronaut on spacecraft)
  • Dilated time (tt) measured by clock in frame moving relative to object (mission control on Earth)
  • Relative motion causes difference between proper and dilated time (time passes slower for moving object)
  • Moving object experiences proper time while stationary observer measures dilated time (particle accelerators)

Twin paradox analysis

  • thought experiment demonstrates time dilation effects (space travel)
  • Traveling twin experiences time dilation due to high-speed motion relative to Earth twin
  • Apparent paradox from situation's symmetry (each twin sees other moving away and returning)
  • Resolved by traveling twin's acceleration and deceleration breaking symmetry (non-inertial motion)
  • Earth twin remains in single while traveling twin changes frames ( applies to inertial frames)
  • Difference in experienced time results from traveling twin's non-inertial motion and relativity of simultaneity ( paths)

Spacetime and Length Contraction

Spacetime concept

  • Spacetime combines three spatial dimensions with time into a four-dimensional continuum
  • Events in spacetime are described by both their position and time coordinates
  • Inertial frames are reference frames moving at constant velocity relative to each other in spacetime

Length contraction

  • occurs for objects moving at high speeds relative to an observer
  • is the length of an object measured in its own rest frame
  • Observed length of a moving object appears shorter than its
  • Length contraction occurs in the direction of motion only

Key Terms to Review (23)

Classical relativity: Classical relativity is the principle that the laws of physics are the same in all inertial frames of reference. It implies that measurements of velocity depend on the relative motion between the observer and what is being observed.
Dilated Time: Dilated time refers to the phenomenon where time appears to slow down or stretch out for an observer who is moving at a high velocity relative to another observer. This concept is a key principle in Einstein's theory of special relativity, which describes the relationship between space, time, and the speed of light.
Einstein synchronization: Einstein synchronization is a method for establishing a coordinate time between two or more observers in different locations, ensuring that their clocks read the same time when signals are exchanged. This concept is crucial in understanding how simultaneity varies depending on the relative motion of observers, particularly in the context of special relativity where the speed of light is constant for all observers. It illustrates that what is simultaneous for one observer may not be so for another moving relative to them, thus laying the groundwork for discussions about time dilation and the nature of time itself.
Inertial Frame: An inertial frame of reference is a coordinate system in which an object with no net force acting on it moves at a constant velocity. It is a frame of reference where Newton's laws of motion hold true and the principle of relativity applies.
Inertial frame of reference: An inertial frame of reference is a frame of reference in which objects not acted upon by forces move in straight lines at constant speeds. It follows Newton's first law of motion, where no acceleration occurs unless acted upon by an external force.
Length contraction: Length contraction is the phenomenon where the length of an object moving at relativistic speeds appears shorter along the direction of motion when observed from a stationary frame of reference. This effect is a direct consequence of Einstein's theory of special relativity.
Length Contraction: Length contraction, also known as Lorentz contraction, is a phenomenon in special relativity where the length of an object measured by an observer moving relative to that object appears to be shorter than its length measured by an observer at rest with respect to the object. This effect is a consequence of the relativity of simultaneity and the constancy of the speed of light.
Light Clock: A light clock is a thought experiment used to illustrate the concept of time dilation in Einstein's theory of special relativity. It consists of a container with a light source and a mirror at each end, where a light pulse is reflected back and forth, and the time it takes for the light to complete one cycle is used to measure time.
Lorentz Factor: The Lorentz factor is a mathematical expression that describes the relationship between the relative speed of an object and the observed effects of special relativity, such as time dilation and length contraction. It is a central concept in Einstein's theory of special relativity and is used to quantify the relativistic changes that occur when an object moves at a significant fraction of the speed of light.
Muon Decay: Muon decay is the radioactive process in which a muon, an unstable subatomic particle, spontaneously transforms into other particles, releasing energy in the form of radiation. This phenomenon is closely related to the concepts of simultaneity and time dilation in the theory of special relativity.
Proper length: Proper length is the length of an object measured by an observer who is at rest relative to the object. It represents the maximum length of the object and is not affected by relative motion.
Proper Length: Proper length, in the context of special relativity, refers to the intrinsic or inherent length of an object as measured by an observer who is at rest with respect to that object. It is the length of an object as determined by a stationary frame of reference, unaffected by the object's motion relative to the observer.
Proper time: Proper time is the time interval measured by an observer who is at rest relative to the event being timed. It is considered the shortest possible time interval between two events.
Proper Time: Proper time refers to the time interval measured by a clock that is at rest relative to the observer. It represents the time elapsed between two events as measured by an observer who is comoving with the system, without any relative motion between the observer and the system.
Reference Frame: A reference frame is a coordinate system used to describe the position, motion, and other physical quantities of an object or event. It provides a frame of reference from which measurements and observations can be made. This term is crucial in understanding various topics in physics, including displacement, one-dimensional kinematics, graphical analysis of motion, and the concepts of simultaneity and time dilation.
Relativity: Relativity is a fundamental concept in physics that describes the relationship between space, time, and the motion of objects. It encompasses two main theories: special relativity and general relativity, which revolutionized our understanding of the universe.
Simultaneity: Simultaneity refers to the occurrence of events at the same time from a specific frame of reference. This concept is crucial in understanding how time is perceived differently depending on the relative motion of observers, which leads to profound implications in physics, especially in terms of how events are synchronized across different frames.
Spacetime: Spacetime is the four-dimensional continuum that combines the three dimensions of space with the one dimension of time into a single framework. This concept revolutionized our understanding of how objects move and interact in the universe, linking space and time in a way that shows they are interdependent rather than separate entities.
Special relativity: Special relativity is a theory formulated by Albert Einstein that describes the physics of objects moving at constant speeds, particularly at speeds close to the speed of light. This theory revolutionized our understanding of space and time, demonstrating that they are interconnected and not absolute. It introduces concepts like time dilation and length contraction, fundamentally altering our perception of motion and the behavior of objects in different frames of reference.
Time dilation: Time dilation is a phenomenon in which the elapsed time between two events is longer for an observer in relative motion compared to an observer at rest. It results from the principles of special relativity, specifically the invariance of the speed of light.
Time Dilation: Time dilation is a fundamental concept in Einstein's theory of special relativity, which states that the passage of time is not absolute but rather depends on the relative motion between an observer and the observed object. This phenomenon occurs when an object moves at a significant fraction of the speed of light, causing time to appear to slow down for that object from the perspective of an observer.
Time-like Interval: A time-like interval is a measure of the separation between two events in spacetime that occurs when the spatial distance between the events is less than the time interval between them. This concept is fundamental in the theory of special relativity and is closely related to the notions of simultaneity and time dilation.
Twin paradox: The twin paradox is a thought experiment in physics that illustrates the effects of time dilation due to relativistic speeds. It involves a scenario where one twin travels at a significant fraction of the speed of light while the other remains on Earth, leading to the traveling twin aging more slowly compared to the stationary twin. This concept highlights how time is not absolute and can vary based on the relative motion between observers, directly tying into the understanding of simultaneity and time dilation.
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28.3 Length Contraction