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- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/06%3A_Vacuum_Solutions/6.E%3A_Vacuum_Solutions_(Exercises)Show that in some other frame such as the rest frame of the sun, in which the planet has speed u toward the incoming probe, the maneuver adds 2u to the speed of the probe. (b) Suppose that we replace ...Show that in some other frame such as the rest frame of the sun, in which the planet has speed u toward the incoming probe, the maneuver adds 2u to the speed of the probe. (b) Suppose that we replace the planet with a black hole, and the space probe with a light ray.
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/05%3A_CurvatureGeneral relativity describes gravitation as a curvature of spacetime, with matter acting as the source of the curvature in the same way that electric charge acts as the source of electric fields. Our ...General relativity describes gravitation as a curvature of spacetime, with matter acting as the source of the curvature in the same way that electric charge acts as the source of electric fields. Our goal is to arrive at Einstein’s field equations, which relate the local intrinsic curvature to the locally ambient matter in the same way that Gauss’s law relates the local divergence of the electric field to the charge density.
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/05%3A_Curvature/5.13%3A_Units_in_General_RelativityAnalyzing units, also known as dimensional analysis, is one of the first things we learn in freshman physics. It’s a useful way of checking our math, and it seems as though it ought to be straightforw...Analyzing units, also known as dimensional analysis, is one of the first things we learn in freshman physics. It’s a useful way of checking our math, and it seems as though it ought to be straightforward to extend the technique to relativity. It certainly can be done, but it isn’t quite as trivial as might be imagined, and it leads to some surprising new physical ideas.
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/03%3A_Differential_Geometry/3.08%3A_The_Metric_in_General_RelativityWhen masses are present, finding the metric is analogous to finding the electric field made by charges, but the interpretation is more difficult. In the electromagnetic case, the field is found on a p...When masses are present, finding the metric is analogous to finding the electric field made by charges, but the interpretation is more difficult. In the electromagnetic case, the field is found on a preexisting background of space and time. In general relativity, there is no preexisting geometry of spacetime. The metric tells us how to find distances in terms of our coordinates, but the coordinates themselves are completely arbitrary.
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/08%3A_Sources
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/10%3A_Appendices/10.03%3A_Appendix_A_(Part_3)To say that a point-event has the X 1 coordinate x 1 means that the projection of the point-event on the axis of X 1 , determined by rigid rods and in accordance with the rules of Euclidean geometry, ...To say that a point-event has the X 1 coordinate x 1 means that the projection of the point-event on the axis of X 1 , determined by rigid rods and in accordance with the rules of Euclidean geometry, is obtained by measuring off a given rod (the unit of length) x 1 times from the origin of coordinates along the axis of X 1 . To say that a point-event has the X 4 coordinate x 4 = t, means that a standard clock, made to measure time in a definite unit period, and which is stationary relative to t…
- 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/Relativity/General_Relativity_(Crowell)/02%3A_Geometry_of_Flat_Spacetime/2.03%3A_Affine_Properties_of_Lorentz_Geometry_(Part_2)For now, let’s refer to a vector such as →e, with the arrow coming in, simply as a “vector,” and the type like c→ as a “dual vector.” In the one-dimensional example of the earth and the cuckoo clock, ...For now, let’s refer to a vector such as →e, with the arrow coming in, simply as a “vector,” and the type like c→ as a “dual vector.” In the one-dimensional example of the earth and the cuckoo clock, the roles played by the two vectors were completely equivalent, and it didn’t matter which one we expressed as a vector and which as a dual vector.
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/10%3A_Appendices/10.01%3A_Appendix_A_(Part_1)From the origin of system k let a ray be emitted at the time \(\tau_{0}\) along the X-axis to x', and at the time \(\tau_{1}\) be reflected thence to the origin of the coordinates, arriving there at t...From the origin of system k let a ray be emitted at the time \(\tau_{0}\) along the X-axis to x', and at the time \(\tau_{1}\) be reflected thence to the origin of the coordinates, arriving there at the time \(\tau_{2}\); we then must have \(\frac{1}{2} (\tau_{0} + \tau_{2}) = \tau_{1}\), or, by inserting the arguments of the function \(\tau\) and applying the principle of the constancy of the velocity of light in the stationary system:—
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/01%3A_Geometric_Theory_of_Spacetime/1.01%3A_Time_and_CausalityYou may have heard that relativity is a theory that can be interpreted using non-Euclidean geometry. The invariance of betweenness is a basic geometrical property that is shared by both Euclidean and ...You may have heard that relativity is a theory that can be interpreted using non-Euclidean geometry. The invariance of betweenness is a basic geometrical property that is shared by both Euclidean and non-Euclidean geometry. We say that they are both ordered geometries. With this geometrical interpretation in mind, it will be useful to think of events not as actual notable occurrences but merely as an ambient sprinkling of points at which things could happen.
- https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/06%3A_Vacuum_Solutions/6.05%3A_Black_Holes_(Part_2)Singularities can also occur without any blow-up in the curvature. An example of this is a conical singularity.
