The most common cases use the x-axis, y-axis, and the line y = x as the line of reflection. There are a number of different types of reflections in the coordinate plane. This is true for any corresponding points on the two triangles and this same concept applies to all 2D shapes. A, B, and C are the same distance from the line of reflection as their corresponding points, D, E, and F. The figure below shows the reflection of triangle ABC across the line of reflection (vertical line shown in blue) to form triangle DEF. The same is true for a 3D object across a plane of refection. In a reflection of a 2D object, each point on the preimage moves the same distance across the line of reflection to form a mirror image of itself. The term "preimage" is used to describe a geometric figure before it has been transformed "image" is used to describe it after it has been transformed. When an object is reflected across a line (or plane) of reflection, the size and shape of the object does not change, only its configuration the objects are therefore congruent before and after the transformation. In geometry, a reflection is a rigid transformation in which an object is mirrored across a line or plane. The assumption of a spacetime admitting a certain symmetry vector field can place restrictions on the spacetime.Home / geometry / transformation / reflection ReflectionĪ reflection is a type of geometric transformation in which a shape is flipped over a line. The Einstein static metric has a Killing algebra of dimension seven (the previous six plus a time translation). For example, the Schwarzschild solution has a Killing algebra of dimension four (three spatial rotational vector fields and a time translation), whereas the Friedmann–Lemaître–Robertson–Walker metric (excluding the Einstein static subcase) has a Killing algebra of dimension six (three translations and three rotations). Generally speaking, the higher the dimension of the algebra of symmetry vector fields on a spacetime, the more symmetry the spacetime admits. For example, Killing vector fields may be used to classify spacetimes, as there is a limit to the number of global, smooth Killing vector fields that a spacetime may possess (the maximum being ten for four-dimensional spacetimes). They also classify spacetimes using symmetry vector fields (especially Killing and homothetic symmetries). Various approaches to classifying spacetimes, including using the Segre classification of the energy–momentum tensor or the Petrov classification of the Weyl tensor have been studied extensively by many researchers, most notably Stephani et al. ( July 2010)Īs mentioned at the start of this article, the main application of these symmetries occur in general relativity, where solutions of Einstein's equations may be classified by imposing some certain symmetries on the spacetime.Ĭlassifying solutions of the EFE constitutes a large part of general relativity research. This statement is equivalent to the more usable condition that the Lie derivative of the tensor under the vector field vanishes: A smooth vector field X on a spacetime M is said to preserve a smooth tensor T on M (or T is invariant under X) if, for each smooth local flow diffeomorphism ϕ t associated with X, the tensors T and ϕ ∗ The implication is that the behavior of objects with extent may not be as manifestly symmetric.) This preserving property of the diffeomorphisms is made precise as follows. (Note that one should emphasize in one's thinking this is a diffeomorphism-a transformation on a differential element. In this approach, the idea is to use (smooth) vector fields whose local flow diffeomorphisms preserve some property of the spacetime. This preservation property which symmetries usually possess (alluded to above) can be used to motivate a useful definition of these symmetries themselves.Ī rigorous definition of symmetries in general relativity has been given by Hall (2004). These and other symmetries will be discussed below in more detail. Symmetries usually require some form of preserving property, the most important of which in general relativity include the following: the Friedmann–Lemaître–Robertson–Walker (FLRW) metric). In cosmological problems, symmetry plays a role in the cosmological principle, which restricts the type of universes that are consistent with large-scale observations (e.g. For example, in the Schwarzschild solution, the role of spherical symmetry is important in deriving the Schwarzschild solution and deducing the physical consequences of this symmetry (such as the nonexistence of gravitational radiation in a spherically pulsating star). Physical problems are often investigated and solved by noticing features which have some form of symmetry. Spacetime symmetries are distinguished from internal symmetries.
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