User:Halibutt/Spacetime/Basic mathematics

Basic mathematics of spacetime summary
^Galilean transformations (click here to return to main)
 * A basic goal is to be able to compare measurements made by observers in relative motion.
 * Transformation between Galilean frames is linear. Given that two coordinate systems are in standard configuration, the coordinate transformation in the x-axis is simply
 * $$x' = x - v t $$


 * Velocities are simply additive. If frame S' is moving at velocity v with respect to frame S, and within frame S', observer O' measures an object moving with velocity u', then
 * $$u' = u - v$$ or  $$u = u' + v$$

^Relativistic composition of velocities (click here to return to main)
 * The relativistic composition of velocities is more complex than the Galilean composition of velocities:
 * $$ u = {v+u'\over 1+(vu'/c^2)} . $$


 * In the low speed limit, the overall result is indistinguishable from the Galilean formula.
 * The sum of two velocities cannot be greater than the speed of light.

^Time dilation and length contraction revisited (click here to return to main)
 * The Lorentz factor, gamma $$\gamma ,$$ appears very frequently in relativity. Given $$\beta = v/c ,$$
 * $$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} $$


 * $$\gamma $$ is the time dilation factor, while $$1 / \gamma $$ is the length contraction factor.
 * The Lorentz factor is undefined for $$ v \ge c .$$

^Lorentz transformations (click here to return to main)
 * The Lorentz transformations combine expressions for time dilation, length contraction, and relativity of simultaneity into a unified set of expressions for mapping measurements between two inertial reference frames.
 * Given two coordinate systems in standard configuration, the transformation equations for the $$t$$ and $$x$$ axes are:
 * $$\begin{align}

t' &= \gamma \left( t - \frac{v x}{c^2} \right) \\ x' &= \gamma \left( x - v t \right) \end{align}$$
 * There have been many alternative derivations of the Lorentz transformations since Einstein's original work in 1905.
 * The Lorentz transformations have a mathematical property called linearity. Because of this: (i) Spacetime looks the same everywhere. (ii) There is no preferred frame. (iii) If two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.

^Doppler effect (click here to return to main)
 * The formulas for classical Doppler effect depend on whether it is the source or the receiver that is moving with respect to the medium.
 * In relativity, there is no distinction between a source moving away from the receiver or a receiver moving away from the source. For the longitudinal Doppler effect, a single formula holds for both scenarios:
 * $$f = \sqrt{\frac{1 - \beta}{1 + \beta}}\,f_0.$$


 * Transverse Doppler shift is a relativistic effect that has no classical analog. Although there are subtleties involved, the basic scenarios can be analyzed by simple time dilation arguments.

^Energy and momentum (click here to return to main)
 * In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector.
 * The relativistic energy-momentum vector has terms for energy and for spatial momentum. Using an uppercase P to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as
 * $$P \equiv (E/c, \vec{p}) = (E/c, p_x, p_y, p_z)$$


 * Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to his famous $$E = m c^2$$ equation as well as to his concept of relativistic mass.

^Conservation laws (click here to return to main)
 * The conservation laws arise from fundamental symmetries of nature.
 * Classical conservation of mass does not hold true in relativity. Since mass and energy are interconvertible, conservation of mass is replaced by conservation of mass-energy.
 * For analysis of energy and momentum problems involving interacting particles, the most convenient frame is usually the "center-of-momentum" frame.
 * Newtonian momenta, calculated as $$p = mv,$$ fail to behave properly under Lorentzian transformation. The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass.

Return to Introduction

Beyond the basics summary
^Rapidity (click here to return to main)
 * The formulas to perform Lorentz transformation and relativistic composition of velocities are nonlinear, making them more complex than the corresponding Galilean formulas. This nonlinearity is an artifact of our choice of parameters.
 * The natural functions for expressing the relationships between different frames are the hyperbolic functions. In a spacetime diagram, the velocity parameter $$\beta$$ is the analog of slope. The rapidity, &phi;, is defined by
 * $$\beta \equiv \tanh \phi \equiv \frac{v}{c}$$


 * Many expressions in special relativity take on a considerably simpler form when expressed in terms of rapidity. For example, the relativistic composition of velocities becomes simply $$\phi = \phi_1 + \phi_2 .$$
 * The Lorentz boost in the x direction becomes a hyperbolic rotation:

\begin{pmatrix} c t' \\ x' \end{pmatrix} = \begin{pmatrix} \cosh \phi & -\sinh \phi \\ -\sinh \phi & \cosh \phi \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix}$$.

^4‑vectors (click here to return to main)
 * General relativity requires knowledge of tensors, which are linear maps between objects like the 4-vectors that belong to the spacetime of relativity. Knowledge of 4-vectors is a prerequisite to understanding tensors.
 * A 4-tuple, A = (A0, A1, A2, A3) is a "4-vector" if its component A&thinsp;i transform between frames according the Lorentz transformation. The last three components of a 4-vector must be a standard vector in three-dimensional space. 4-vectors exhibit closure under linear combination, inner-product invariance, and invariance of the magnitude of a vector.
 * Examples of 4-vectors include the displacement 4-vector, the velocity 4-vector, the energy-momentum 4-vector, and the acceleration 4-vector.
 * The use of momentarily comoving reference frames enables special relativity to deal with accelerating particles.
 * Physical laws must be valid in all frames, but the laws of classical mechanics with their time-dependent 3-vectors fail to behave properly under Lorentz transformation. Valid physical laws must be formulated as equations connecting objects from spacetime like scalars and 4-vectors via tensors of suitable rank.

^Acceleration (click here to return to main)
 * It is a common misconception that special relativity is unable to handle accelerating objects or accelerating reference frames. Special relativity handles such situations quite well. It is only when gravitation is significant that general relativity is required.
 * The Dewan–Beran–Bell spaceship paradox is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues. The issues become almost trivial when analyzed with the aid of spacetime diagrams.
 * Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons.

Return to Introduction

Introduction to curved spacetime summary
^Basic propositions (click here to return to main)
 * General relativity asserts that "action-at-a-distance" does not exist. The motions of a satellite orbiting the Earth are not dictated by distant forces exerted by the Earth, Moon and Sun. Rather, the satellite is always following a straight line in its local inertial frame.
 * Each particle's local frame varies from point to point as a result of the curvature of spacetime.
 * General relativity is based on two central propositions: (1) The laws of physics cannot depend on what coordinate system one uses. (2) In any sufficiently small region of space, the effects of gravitation are the same as those from acceleration. This second proposition is the equivalence principle.

^Curvature of time (click here to return to main)
 * Gravitational fields make it impossible to construct a global inertial frame, as is required by special relativity.
 * A photon climbing in Earth's gravitational field loses energy and is redshifted.
 * The gravitational redshift implies that gravity makes time run slower. This amounts to a statement that time is curved.
 * The prediction of curved time is not unique to general relativity. Rather, it is predicted by any theory of gravitation that respects the principle of equivalence.
 * Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space.

^Curvature of space (click here to return to main)
 * Curvature of time completely accounts for all Newtonian gravitational effects.
 * There are curvature terms for the spatial components of the invariant interval as well, but the effects on planetary orbits and the like are tiny. This is because the speeds of planets and satellites in their orbits are very much slower than the speed of light.
 * Nevertheless, Urbain Le Verrier, in 1859, was able to demonstrate discrepancies in the orbit of Mercury from that predicted by Newton's laws.
 * Einstein showed that this discrepancy, the anomalous precession of Mercury, is explained by the spatial terms in the curvature of spacetime.
 * For light, the spatial terms in the invariant interval are comparable in magnitude to the temporal term, so the effects of the curvature of space are comparable to the effects of the curvature of time.
 * The famous 1919 Eddington eclipse expedition showed that the bending of light around the Sun includes a component explained by the curvature of space.

^Sources of spacetime curvature (click here to return to main)


 * In Newton's theory of gravitation, the only source of gravitational force is mass. In contrast, general relativity identifies several sources of spacetime curvature in addition to mass: Mass-energy density, momentum density, pressure, and shear stress.
 * Gravity itself is a source of gravity.
 * Moving or rotating masses can generate gravitomagnetic fields analogous to the magnetic fields generated by moving charges.
 * Pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation.
 * Experiment has verified the ability of pressure and momentum to act as sources of spacetime curvature. Only stress has eluded experimental verification as a source of spacetime curvature, although mathematical consistency of the Einstein field equations demands that it acts so.

Return to Introduction