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- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/03%3A_Kinematics/3.02%3A_The_stretch_factor_is_the_Doppler_shiftThe stretching and squishing factors for the diagonals are the same as the Doppler shift. We notate this factor as D (which can stand for either “Doppler” or “diagonal”).
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/06%3A_Waves/6.05%3A__The_Doppler_Shift_and_AberrationWe generalize our previous discussion of the Doppler shift of light to 3+1 dimensions. Imagine that rain is falling vertically while you drive in a convertible with the top down. To you, the raindro...We generalize our previous discussion of the Doppler shift of light to 3+1 dimensions. Imagine that rain is falling vertically while you drive in a convertible with the top down. To you, the raindrops appear to be moving at some nonzero angle relative to vertical. This is referred to as aberration.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/03%3A_Kinematics/3.01%3A_How_can_they_both_._._._%3FIt’s neither helpful nor necessary to break down the observations into a factor describing what “really” happens and a correction factor to account for the relativistic distortions of “reality.” All w...It’s neither helpful nor necessary to break down the observations into a factor describing what “really” happens and a correction factor to account for the relativistic distortions of “reality.” All we need to worry about is the world-lines and intersections of worldlines shown in the spacetime diagrams, along with the metric, which allows us to compute how much proper time is experienced by each observer.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/03%3A_Kinematics/3.07%3A_The_Projection_OperatorA frequent source of confusion in relativity is that we write down equations that are coordinate-dependent, but forget the dependency. Similarly, it is possible to write expressions that are only vali...A frequent source of confusion in relativity is that we write down equations that are coordinate-dependent, but forget the dependency. Similarly, it is possible to write expressions that are only valid for one choice of signature. The following notation, defining a projection operator P , is one tool for avoiding these difficulties.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/02%3A_Foundations/2.02%3A_FlatnessEuclidean geometry is only an approximate description of the earth’s surface, for example, and this is why flat maps always entail distortions of the actual shapes. The distortions might be negligible...Euclidean geometry is only an approximate description of the earth’s surface, for example, and this is why flat maps always entail distortions of the actual shapes. The distortions might be negligible on a map of Connecticut, but severe for a map of the whole world. That is, the globe is only locally Euclidean. On a spherical surface, the appropriate object to play the role of a “line” is a great circle. The lines of longitude are examples of great circles.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/02%3A_Foundations/2.05%3A_Lemma_-_Spacetime_area_is_invariantThe area in the x−t plane is invariant, i.e., it does not change between frames of reference.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/03%3A_Kinematics/3.03%3A_Combination_of_VelocitiesIn nonrelativistic physics, velocities add in relative motion. For example, if a boat moves relative to a river, and the river moves relative to the land, then the boat’s velocity relative to the land...In nonrelativistic physics, velocities add in relative motion. For example, if a boat moves relative to a river, and the river moves relative to the land, then the boat’s velocity relative to the land is found by vector addition. This linear behavior cannot hold relativistically.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/06%3A_Waves/6.07%3A_Abstract_Index_NotationIn abstract index notation, however, the a is simply a name we gave to a pipe coming into vector o; the fact that we didn’t need to refer to the name in order to connect it to some other pipe is i...In abstract index notation, however, the a is simply a name we gave to a pipe coming into vector o; the fact that we didn’t need to refer to the name in order to connect it to some other pipe is irrelevant. For reasons to be explained in section 7.4, a partial derivative with respect to a coordinate, such as ∂/∂xk, is treated as if the index were a subscript, and conversely ∂/∂xk is considered to have a superscripted k.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/06%3A_Waves/6.06%3A_Phase_and_Group_VelocityWe could try to resolve the ambiguity by requiring that the arrow’s projection into the xy plane be perpendicular to the intersection of the wavefronts with that plane, but (with the exception of ...We could try to resolve the ambiguity by requiring that the arrow’s projection into the xy plane be perpendicular to the intersection of the wavefronts with that plane, but (with the exception of the case where the wave travels at c, Example \PageIndex1) this prescription gives results that change depending on our frame of reference, and the changes are not describable by a Lorentz transformation of the velocity vector.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/09%3A_Flux/9.03%3A_Gauss%E2%80%99s_TheoremThe connection between the local and global conservation laws is provided by a theorem called Gauss’s theorem. In your course on electromagnetism, you learned Gauss’s law, which relates the electric f...The connection between the local and global conservation laws is provided by a theorem called Gauss’s theorem. In your course on electromagnetism, you learned Gauss’s law, which relates the electric flux through a closed surface to the charge contained inside the surface. In the case where no charges are present, it says that the flux through such a surface cancels out.
- https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/01%3A_Spacetime/1.02%3A_Minkowski_CoordinatesIt is often convenient to name points in spacetime using coordinates, and a particular type of naming, chosen by Einstein and Minkowski, is the default in special relativity. I’ll refer to the coordin...It is often convenient to name points in spacetime using coordinates, and a particular type of naming, chosen by Einstein and Minkowski, is the default in special relativity. I’ll refer to the coordinates of this system as Minkowski coordinates, and they’re what I have in mind throughout this book when I use letters like t and x without further explanation.
