Tensors in Relativity

from my book “the geometry of curvature and gravitation waves” www.mpantes .gr definition the sufficient and necessary condition for a space to be flat is that the Riemann tensor is zeroed. The curvature of the surface, and later in 4-dimensional space-time, interfering with the metric tensor, gave a new type of distance, angle, yet a new definition of parallelism, something that took us away from Euclidean geometry. 5.1 first we have the relations of of connecting tensors 𝑹𝒑𝒓𝒔𝒕=𝒈𝒑𝒎 𝑹𝒎 .𝒓𝒔𝒕 𝑹𝒋𝒌= 𝒈𝒊𝒑 𝑹𝒋𝒊𝒑𝒌 5.2.2 relations with the second covariant derivate (1.20) X k ,ij  X k , ji   Rijlk X l l X kl , i j  X kl , j i  Rijkm X ml  Rijm X km X kl , i j  X kl , j i  Rijkm X ml  Rijlm X km k l X k, il j  X k, jli   Rijm X m l  Rijm X km 𝒑 𝒑 relations of anti-symmetry 𝑹𝒓𝒔𝒕 = −𝑹𝒓𝒔𝒕 𝑹𝒑𝒓𝒔𝒕 = −𝑹𝒓𝒑𝒔𝒕 𝑹𝒑𝒓𝒔𝒕 = −𝑹𝒑𝒓𝒕𝒔 𝑹𝒑𝒓𝒔𝒕 = 𝑹𝒔𝒕𝒑𝒓 so R prst is antisymmetric as p,r and also as s,t but symmetric as the two pairs of under indices relations of symmetry Rijkl  R jkil  Rkijl  0 R jkli  Rklji  Rljki  0 Rklij  Rlikj  Riklj  0 Rlijk  Rijlk  R jlik  0 Let's go back to the needs of formalism. If we want to describe the motion of a body, it is not enough to consider only its orbit, for a complete description, but also its speed. Inertial motion is not only linear but also uniform. Here time enters on an equal footing with space, three coordinates for space and another dimension for time to have a complete description of motion, in a complete geometric form. So we need three dimensions to describe the trajectory of a motion and four dimensions to give a complete description of the motion. The new equation of straight line of Euclidean form, now is given by the geodesic, the equation of the straight line in general curvilinear coordinates.  d 2u a du  a du   0  ds ds ds 2 if the index values develop at 0,1,2,3 and with the summation condition, we have the geodesic in curved space-time. We can understand these equations in another way by observing that in Cartesian coordinates, the symbols of Christopher are all zero goes back to d 2u a 0 ds 2 which are the equations of a straight line in such coordinates. i.e. at the plane, otherwise they cease to be the known straight lines .. in the curved space there are no Euclidean straight lines. 5.4 Ο τανυστής Ricci in general relativity theory we use the tensor2 του Ricci 𝑹𝒊𝒋 , through through of 𝑹𝒊𝒋 = 𝒈𝒔𝒏 𝑹𝒔𝒊𝒋𝒏 . Ρήμαν tensor , (5.4.1) from 5..2.4 we have the idiot of symmetry 𝑹𝒊𝒋 = 𝒈𝒔𝒏 𝑹𝒔𝒊𝒋𝒏 = 𝒈𝒔𝒏 𝑹𝒋𝒏𝒔𝒊 = 𝑹𝒋𝒊 This is the concept of the four-dimensional space-time, and we saw a form of it in equation 1.32 for the two-dimensional curved surface of the surfaces. In Physics, they represent orbits of free-moving particles in which no forces are exerted in space-time. When it is flat, the geodetic capsules are straight and this is easily ascertained by Christoffel's Jedi. In a curved area the Christoffel symbols are not and consequently the geodesic cease to be straight. But is the four-dimensional description real? Is there space-time? Minkowski, a devoted devotee of pure mathematics, adopted the view that the four-dimensional curve exists "really" and that the various descriptions are different representations of the same reality. He showed that the various descriptions of a movement represented by a curve within the four-dimensional space are projections of this curve in different threedimensional spaces. For Minkowski the real thing is "the simplest theoretical representation of our experiments" while on the other hand body is real- the word means “our experience expressed as accurately as possible in our usual everyday language. The Newtonian space-time consists of the three-dimensional Euclidean space and time, during which the former remains unchanged (the Newtonian space-time is mathematically represented as R3xR). There is no metric in Newton's space-time as multiplicity, to measure the "distance of two events, one in Thessaloniki today and the other in Athens tomorrow. Here the Euclidean space has a metric and time is measured separately by a global clock. 5.5 𝑹𝒑𝒓𝒔𝒕=𝒈𝒑𝒎 𝑹𝒎 .𝒓𝒔𝒕 in principle, the relations of the connected tensors apply 𝑹𝒋𝒌= 𝒈𝒊𝒑 𝑹𝒋𝒊𝒑𝒌 5.6 relations with the second covariant derivative (1.20) X k ,ij  X k , ji   Rijlk X l l X kl , i j  X kl , j i  Rijkm X ml  Rijm X km X kl , i j  X kl , j i  Rijkm X ml  Rijlm X km k l X k, il j  X k, jli   Rijm X m l  Rijm X km 𝒑 𝒑 .7 antisymmetry relations 𝑹𝒓𝒔𝒕 = −𝑹𝒓𝒔𝒕 𝑹𝒑𝒓𝒔𝒕 = −𝑹𝒓𝒑𝒔𝒕 𝑹𝒑𝒓𝒔𝒕 = −𝑹𝒑𝒓𝒕𝒔 𝑹𝒑𝒓𝒔𝒕 = 𝑹𝒔𝒕𝒑𝒓 5.8 symmetry relations Rijkl  R jkil  Rkijl  0 R jkli  Rklji  Rljki  0 Rklij  Rlikj  Riklj  0 Rlijk  Rijlk  R jlik  0 Minkowski's space-time relationship seems to be the mathematical model of a real four-dimensional world. But what does the curved spacetime mean? It means that as the surface is curved during the Third dimension, our known space is curved during the fourth dimension! But a surface or a line can be curved in space, but to say that the space itself is curved seems incalculable and absurd. But it is futile to try to portray what a curved space looks like, we can only describe it by measuring the triangles. The image of curved surfaces is a "intuitive ratio" a guide to the mathematical course of models. So what mathematicians mean and understand in terms of curvature is not what the word conveys in everyday speech, here it means that we measure distances differently, and the relationships between the distances of points are different from the relationships of Euclidean geometry. The curvature has nothing to do with the shape of the space whether it is bent or not - but it is defined exclusively by the metric. it is not the space that is curved but the geometry of the space. Although curved space has been studied by pure mathematics for almost a century, it was Einstein in 1915 who discovered curvature in real space around us. This is because for mathematicians there is no commitment to consider Euclidean or non-Euclidean metric (mathematical space), but physicists must set the distance in accordance with the observed properties of the physical distance eg the time it takes for a light signal to travel. a distance (physical space). END OF THE RIEMMAN GEOMETRY IN RELATIVITY TENSORS IN N DIMENSIONS Tensors in N dimensions what is the Riemann tensor in N dimensional space? It is the symbol here in four dimenstions p R.rst    p rt  t rsp  rtm msp  rsm mtp s x x 5.10 The Riemann’s tensor in space-time the theory of tensors is the basic mathematical tool of the theory of relativity and especially of the general theory. A particularly useful point in the analysis of the curvature tensor in genegal relativity is Bianchi’s identities, with are always satisfied by the Riemann tensor and the fact that the metric tensor has zero covariant derivative. the identities with a role in general relativity are 𝑹𝒓𝒔𝒎𝒏,𝒕 + 𝑹𝒓𝒔𝒏𝒕,𝒎 + 𝑹𝒓𝒔𝒕𝒎,𝒏 = 𝟎 We approach here the tensor of Riemann on curved surfaces, for simplicity that defines the curvature. In surfaces for simplicity, the procedure is the concept of parallelism field of vectors in each point of a curve on a surface. If we are given a circuit on the surface and, starting with a given vector at a point of the circuit, we take the parallel vectors along the curve with respect to the surface and this vector field is defined from  dX   du a   X  0) dt dt This is our definition of parallel vectors with respect to the surface and it will be noticed that surface vectors are defined as parallel along a curve on the surface. Then there is not a priori reason why we should arrive at the same initial vector when we ha completed the circuit.( Tullio Levi-Civita). In fact the parallel propagation with respect to the surface of a vector round a closed circuit generally results in a new vector when we have again arrived at the initial point. It is proved that the vector changes, by an amount proportional to a tensor of fourth order, of the tensor of curvature, the Riemann tensor. So we have a definition with the geometric picture, without the algebraic proofs that alienate us. What is the curvature tensor of the surface? It is the symbol here in four dimenstions p R.rst    p rt  t rsp  rtm msp  rsm mtp s x x the Riemann tensor satisfies a multitude of important symmetries, which derive from its close relationship with exchange production. that gives a set of numbers that Riemann introduced for each point in space, which would describe how curved he was. He found that in 4 dimensions 44 numbers are needed at each point to fully describe the properties of a multiplicity, no matter how distorted it may be. As the symmetries significantly reduce the number of independent components that ultimately for a 4-dimensional spacecraft take only 20 A combination of metric tensor values gives us the curvature of space. Since Riemann's tensor is expressed only as a function of the metric tensor gik and its derivatives, when the space is Euclidean, there is a reference system in which the gik’ are constant and the curvature tensor is zero. And it is known that when the components of a tensor are zeroed in a reference system, then they are zeroed in any applicable system, thus playing a key role in the n-dimensional differential geometry, since it categorizes a group of spaces. For n = 2 these spaces are called expandable surfaces and are for example the plane, the cylinder, the cone, etc. In non-Euclidean spaces, on the other hand, we cannot define Cartesian reference systems. So we have the following definition Definition If in space there is a reference system in which the components of the metric tensor are constant, the space is called flat. It turns out that the sufficient and necessary condition for a space to be flat is that the tensor Riemann is zeroed. 5.11 Example. We have seen that in cylindrical coordinates that develop on the surface of a right cylinder the metric tensor is 2 ( g ik )    0 0  1  ρ=constant. So the right cylindrical surface is a flat space and its Riemann is zero. l X k ,ij  X k , ji  Rijk Xl R.rps t  R.sp t r  R.pt r s  0 R p r s t   R r p s t antisymmetry relations R p r s t   R P r t s ...................................................................................................................(1.42) R p r s t  Rs t p r 512 Bianchi identity Tensor theory is the basic mathematical tool of relativity theory and especially general theory. A particularly useful point of analysis of the curvature tensor in general relativity is the Bianchi identity. It is 𝑹𝒓𝒔𝒎𝒏,𝒕 + 𝑹𝒓𝒔𝒏𝒕,𝒎 + 𝑹𝒓𝒔𝒕𝒎,𝒏 = 𝟎 5.13 Ricci tensor From the Riemann tensor we define the Ricci tensor contractions as via Rjk=gsn Rsijn . Following the symmetry property we haveR mk =Rkm In the general theory of relativity and in other physical and geometric applications we use the Ricci tensor with components symbolized by the symbol. It comes from the contraction of the first and third index of the stretcher Riemann. Because shrinkage can also occur in the other indicators of the curvature tensor we have the following clarifications j j  R  Rik ....................................................................................(1.45) R i jk jik Rikj k  0 Rii j k  Rii k j another additional contraction of RiJ gives the invariant curvature Because it is antisymmetric with respect to i, j, and k, l all other contractions are either zeroed or reduced to So the tensor Ricci is essentially the only contraction of the Riemann tensor 5.14The Einstein tensor. We consider the mixed tensor 1 G ij  R ij   ij R 2 This is the tensor of Einstein . Its covariant and contravariant components are respectively Rij  1 g ij R, 2 R ij  1 ij g R 2 Its divergence is (from 1.54) 1 R 1 R 0 G ij ,i  R ij ,i   ij i  R ij ,i  2 x 2 x j If we apply contraction, we achieve the dimension of space, which does not change with N. This tensor was named because Einstein was the first to recognize its importance in gravity. in the fourth dimension of curved space time, .  d 2u a du  a du   0  ds ds ds 2 d 2u a 0 ds 2 du  dX   a X   0) dt dt p R.rst    p rt  t rsp  rtm msp  rsm mtp s x x 2 ( g ik )    0 0  1  l X k ,ij  X k , ji  Rijk Xl R.rps t  R.sp t r  R.pt r s  0 R p r s t   Rr p s t R p r s t   R P r t s ...................................................................................................................(1.42) R p r s t  Rs t p r 𝒓 𝑹𝒓𝒔𝒎𝒏,𝒕 + 𝑹𝒓𝒔𝒏𝒕,𝒎 + 𝑹𝒔𝒕𝒎,𝒏 𝑹𝒓𝒔𝒎𝒏,𝒕 + 𝑹𝒓𝒔𝒏𝒕,𝒎 + 𝑹𝒓𝒔𝒕𝒎,𝒏 = 𝟎 j j  R  Rik ....................................................................................(1.45) R i jk jik Rikj k  0 Rii j k  Rii k j 1 G ij  R ij   ij R 2 Rij  1 g ij R, 2 R ij  1 ij g R 2 1 R 1 R G ij ,i  R ij ,i   ij i  R ij ,i  0 2 x 2 x j 𝑻′ 𝝁𝝂 𝝏𝒙′𝝁 𝝏𝒙′𝝂 =𝑻 … … … … … … … .4.8) 𝝏𝒙𝒌 𝝏𝒙𝒍 𝒌𝒍 𝑑 𝑑𝑡 𝑑 𝑑𝑡 𝜕𝐿 𝜕𝑞 𝜄 = 𝑑𝑝 𝑖 𝑑𝑡 𝜕𝐿 𝜕𝑞 𝜄 = =0 𝜕𝜈 𝛵𝜇𝜈 = 0 𝜕0 𝛵 0𝑎 = 0 𝜕𝐿 𝜕𝑞 𝑖 𝜕𝐿 𝜕𝑞 𝜄 = 𝑝𝑖 𝛵 𝝁𝝂 𝜇𝜈 ,𝜈 =0 𝝁𝝂 𝑻;𝝂 = 𝝏𝝂 𝜯 … … … … … … … … 4.9) 𝝁 + 𝜞𝜶𝝂 𝜯𝜶𝝂 + 𝜞𝝂𝜶𝝂 𝜯𝜶𝝁 … … … … . (4.10) GENERAL RELATIVITY: the equivalence principle The principle of equivalence