The geometry of space-time
Main article: Minkowski space In physics and mathematics, Minkowski space is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime. Minkowski space is namedSR uses a 'flat' 4-dimensional Minkowski space, which is an example of a space-time In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions. According to certain Euclidean space perceptions, the universe has three. This space, however, is very similar to the standard 3 dimensional Euclidean space Around 300 BC, the Greek mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space. An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. Today these relationships are known as two- and three-dimensional Euclidean geometry, and fortunately by that fact, very easy to work with.
The differential In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable, then a change in the value of x is often denoted Δx . The differential dx represents such a change, but is infinitely small. Although, as stated, it is not a precise mathematical concept, it is extremely useful intuitively, of distance (ds) in cartesian 3D space is defined as:
where (dx1,dx2,dx3) are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension is added, derived from time, so that the equation for the differential of distance becomes:
- .
If we wished to make the time coordinate look like the space coordinates, we could treat time as imaginary An imaginary number, in mathematics, is a number in the form bi where b is a real number and i is the square root of minus one, known as the imaginary unit. Imaginary numbers and real numbers may be combined as complex numbers in the form a + bi where a is the real part and bi is the imaginary part. Imaginary numbers can therefore be thought of as: x4 = ict . In this case the above equation becomes symmetric:
- .
This suggests what is in fact a profound theoretical insight as it shows that special relativity is simply a rotational symmetry Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has three rotational symmetries (or "a threefold of our space-time In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions. According to certain Euclidean space perceptions, the universe has three, very similar to rotational symmetry of Euclidean space Around 300 BC, the Greek mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space. An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. Today these relationships are known as two- and three-dimensional Euclidean geometry. Just as Euclidean space uses a Euclidean metric In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. By using this formula as distance, Euclidean space becomes a metric space . The associated norm is called the Euclidean norm, so space-time uses a Minkowski metric. Basically, SR can be stated in terms of the invariance of space-time interval (between any two events) as seen from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime. It is a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer of a point. That is, the full Poincaré group is the affine group) of Minkowski space-time. According to Misner (1971 §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) rather than a "disguised" Euclidean metric using ict as the time coordinate.
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space
- ,
we see that the null In general relativity, geodesics generalize the notion of "straight lines" to curved spacetime. This concept is based on the mathematical concept of a geodesic. Importantly, the world line of a particle free from all external force is a particular type of geodesic. In other words, a freely moving particle always moves along a geodesic geodesics In mathematics, a geodesic /ˌdʒiəˈdɛsɪk, -ˈdisɪk/[jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a metric, geodesics are defined to be the shortest path between points on the space. In the presence of an affine connection, geodesics are defined to lie along a dual-cone:
defined by the equation
or simply
— which is the equation of a circle with r=c×dt. If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:
- .
This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars A star is a massive, luminous ball of plasma that is held together by gravity. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth. Other stars are visible in the night sky, when they are not outshone by the Sun. Historically, the most prominent stars on the celestial sphere were grouped together into and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance away and a time d/c in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)
The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.
The geometry of Minkowski space can be depicted using Minkowski diagrams The Minkowski diagram was developed in 1908 by Herman Minkowski and provides an illustration of the properties of space and time in the special theory of relativity. It allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations, which are useful also in understanding many of the thought-experiments in special relativity.
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