Introduction to general relativity

context: en: A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic latitude and longitude. Unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,100 mi), while for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,200 mi). Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface (or space, or spacetime) and relates coordinate differences to differences in distance. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function.The metric function and its rate of change from point to point can be used to define a geometrical quantity called the Riemann curvature tensor, which describes exactly how the space or spacetime is curved at each point. In general relativity, the metric and the Riemann curvature tensor are quantities defined at each point in spacetime. As has already been mentioned, the matter content of the spacetime defines another quantity, the energy–momentum tensor T, and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity G, now called the Einstein tensor, which describes some aspects of the way spacetime is curved. Einstein's equation then states that
qas: question: zh_tw: 什麼型別的簡單表面無法用座標描述？
answers: en: text: spherical

context: en: One of the most important aspects of general relativity is that it can be applied to the universe as a whole. A key point is that, on large scales, our universe appears to be constructed along very simple lines: all current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the universe is approximately homogeneous and isotropic. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current cosmological models of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's matter content, namely thermodynamics, nuclear- and particle physics. According to these models, our present universe emerged from an extremely dense high-temperature state – the Big Bang – roughly 14 billion years ago and has been expanding ever since.Einstein's equations can be generalized by adding a term called the cosmological constant.  When this term is present, empty space itself acts as a source of attractive (or, less commonly, repulsive) gravity. Einstein originally introduced this term in his pioneering 1917 paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for constructing static model universes within the framework of general relativity. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term. Since the end of the 1990s, however, astronomical evidence indicating an accelerating expansion consistent with a cosmological constant – or, equivalently, with a particular and ubiquitous kind of dark energy – has steadily been accumulating.
qas: question: zh_tw: 怎樣的高密度和極高溫的狀態，開始了現知的宇宙？
answers: en: text: Big Bang

context: en: The discoverers of PSR1913+16, Russell Hulse and Joseph Taylor, were awarded the Nobel Prize in Physics in 1993. Since then, several other binary pulsars have been found. The most useful are those in which both stars are pulsars, since they provide accurate tests of general relativity.Currently, a number of land-based gravitational wave detectors are in operation, and a mission to launch a space-based detector, LISA, is currently under development, with a precursor mission (LISA Pathfinder) which was launched in 2015. Gravitational wave observations can be used to obtain information about compact objects such as neutron stars and black holes, and also to probe the state of the early universe fractions of a second after the Big Bang.
qas: question: zh_tw: 地心引力波可以用來幹什麼？
answers: en: text: to obtain information about compact objects

context: en: The equivalence between mass and energy, as expressed by the formula E = mc2, is the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its temperature or the binding energy of systems such as nuclei or molecules, contribute to that body's mass, and hence act as sources of gravity.In special relativity, energy is closely connected to momentum. Just as space and time are, in that theory, different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call four-momentum. In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal pressure and tension. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the energy–momentum tensor.
qas: question: zh_tw: 五種引力的來源是什麼？
answers: en: text: mass, energy, momentum, pressure and tension

context: en: The first new effect is the gravitational frequency shift of light. Consider two observers aboard an accelerating rocket-ship.  Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and unattached objects accelerate in the opposite direction, falling "downward". Assume that one of the observers is "higher up" than the other. When the lower observer sends a light signal to the higher observer, the acceleration causes the light to be red-shifted, as may be calculated from special relativity; the second observer will measure a lower frequency for the light than the first. Conversely, light sent from the higher observer to the lower is blue-shifted, that is, shifted towards higher frequencies. Einstein argued that such frequency shifts must also be observed in a gravitational field. This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described below.
qas: question: zh_tw: 根據愛因斯坦的說法，在哪裡可以觀察到光波向較高頻率的移動？
answers: en: text: gravitational field