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- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/01%3A_Geometric_Theory_of_Spacetime/1.03%3A_Non-simultaneity_and_Maximum_Speed_of_Cause_and_EffectInstantaneous communication is impossible. There must be some maximum speed at which signals can propagate — or, more generally, a maximum speed at which cause and effect can propagate — and this spee...Instantaneous communication is impossible. There must be some maximum speed at which signals can propagate — or, more generally, a maximum speed at which cause and effect can propagate — and this speed must for example be greater than or equal to the speed at which radio waves propagate. It is also evident from these considerations that simultaneity itself cannot be a meaningful concept in relativity.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/15%3A_Special_Relativity/15.11%3A_The_Twins_ParadoxDuring the late 1950s and early 1960s there was great controversy over a problem known as the “Twins Paradox”. The controversy was not confined to within scientific circles, but was argued, by scienti...During the late 1950s and early 1960s there was great controversy over a problem known as the “Twins Paradox”. The controversy was not confined to within scientific circles, but was argued, by scientists and others, in the newspapers, magazines and many serious journals. . The paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Big_Ideas_in_Cosmology_(Coble_et_al.)/09%3A_Special_Relativity/9.05%3A_Applications_of_SpacetimeYou will explore aspects of the feasibility of interstellar travel. You will understand that events that are simultaneous in one frame are not necessarily simultaneous in another. You will understand ...You will explore aspects of the feasibility of interstellar travel. You will understand that events that are simultaneous in one frame are not necessarily simultaneous in another. You will understand that events are real, that the same events happen in both frames, but that observers in different frames disagree about the spacetime coordinates of the events.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/17%3A_Relativistic_Mechanics/17.03%3A_Special_Theory_of_RelativityEinstein's Special Theory of Relativity.
- https://phys.libretexts.org/Bookshelves/Relativity/Book%3A_Relativity_Lite_-_A_Pictorial_Translation_of_Einsteins_Theories_of_Motion_and_Gravity_(Straton)/01%3A_Chapters/1.03%3A_A_Trip_to_Alpha_CentauriThis is quite close to the 1.22 overall time-dilation factor for the trip given by our crude counting of rings on a grid. (As with the earlier trip using instantaneous jumps from stillness to motion, ...This is quite close to the 1.22 overall time-dilation factor for the trip given by our crude counting of rings on a grid. (As with the earlier trip using instantaneous jumps from stillness to motion, and the reverse, the initial acceleration-induced time effects at the beginning of the trip and the final deceleration-induced time effects at the end of the trip will essentially cancel the time-dilation effects of the unaccelerated portion of the trip.)
- https://phys.libretexts.org/Bookshelves/Relativity/Spacetime_Physics_(Taylor_and_Wheeler)/04%3A_Trip_to_Canopus/4.06%3A_Twin_Paradox\[\left(\begin{array}{c} \text { Proper time: } \\ \text { Earth to } \\ \text { Canopus } \end{array}\right)^{2}=\left(\begin{array}{c} \text { rocket time: } \\ \text { Earth to } \\ \text { Canopus...\[\left(\begin{array}{c} \text { Proper time: } \\ \text { Earth to } \\ \text { Canopus } \end{array}\right)^{2}=\left(\begin{array}{c} \text { rocket time: } \\ \text { Earth to } \\ \text { Canopus } \end{array}\right)^{2}=\left(\begin{array}{c} \text { Earth time: } \\ \text { Earth to } \\ \text { Canopus } \end{array}\right)^{2}-\left(\begin{array}{c} \text { Earth distance: } \\ \text { Earth to } \\ \text { Canopus } \end{array}\right)^{2} \nonumber\]
- https://phys.libretexts.org/Courses/Grand_Rapids_Community_College/PH246_Calculus_Physics_II_(2025)/14%3A__Relativity/14.06%3A_The_Lorentz_TransformationRelativistic phenomena can be explained in terms of the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes. The analysis of relativ...Relativistic phenomena can be explained in terms of the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes. The analysis of relativistic phenomena in terms of space-time diagrams supports the conclusion that these phenomena result from properties of space and time itself, rather than from the laws of electromagnetism.
- https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/24%3A_The_Theory_of_Special_Relativity/24.03%3A_Time_DilationThe muon is traveling with a speed of \(v=0.9c\) relative to the Earth, thus the gamma factor is given by: \[\begin{aligned} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} =\frac{1}{\sqrt{1-0.9^2}}=2.29\...The muon is traveling with a speed of \(v=0.9c\) relative to the Earth, thus the gamma factor is given by: \[\begin{aligned} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} =\frac{1}{\sqrt{1-0.9^2}}=2.29\end{aligned}\] The amount of time that goes by in the frame of reference of the Earth, \(\Delta t\), when \(\Delta t'=2.2\mu\text{s}\) has gone by in the muon’s frame of reference will be dilated by the gamma factor. \(\Delta t'\) is the proper time in the muon frame’s of reference, which correspon…
- https://phys.libretexts.org/Courses/Berea_College/Introductory_Physics%3A_Berea_College/24%3A_The_Theory_of_Special_Relativity/24.03%3A_Time_DilationThe muon is traveling with a speed of \(v=0.9c\) relative to the Earth, thus the gamma factor is given by: \[\begin{aligned} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} =\frac{1}{\sqrt{1-0.9^2}}=2.29\...The muon is traveling with a speed of \(v=0.9c\) relative to the Earth, thus the gamma factor is given by: \[\begin{aligned} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} =\frac{1}{\sqrt{1-0.9^2}}=2.29\end{aligned}\] The amount of time that goes by in the frame of reference of the Earth, \(\Delta t\), when \(\Delta t'=2.2\mu\text{s}\) has gone by in the muon’s frame of reference will be dilated by the gamma factor. \(\Delta t'\) is the proper time in the muon frame’s of reference, which correspon…
- https://phys.libretexts.org/Courses/Muhlenberg_College/MC_%3A_Physics_213_-_Modern_Physics/01%3A__Relativity/1.06%3A_The_Lorentz_TransformationRelativistic phenomena can be explained in terms of the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes. The analysis of relativ...Relativistic phenomena can be explained in terms of the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes. The analysis of relativistic phenomena in terms of space-time diagrams supports the conclusion that these phenomena result from properties of space and time itself, rather than from the laws of electromagnetism.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/01%3A_Spacetime/1.01%3A_Three_Models_of_SpacetimeTime and space together make spacetime, the stage on which physics is played out. Until 1905, physicists were trained to accept two mutually contradictory theories of spacetime. I’ll call these the Ar...Time and space together make spacetime, the stage on which physics is played out. Until 1905, physicists were trained to accept two mutually contradictory theories of spacetime. I’ll call these the Aristotelian and Galilean views, although my colleagues from that era would have been offended to be accused of even partial Aristotelianism.