FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W3148758246> ?p ?o ?g. }

• W3148758246 abstract "I t is shown that, also in the mixed initial and boundary value problem, Einstein's equations may be replaced by the two subsystems T,~//,~ = 0 and R ~ = -~ T ~ -~. T g ~ , provided that the initial data verify the consistency conditions G~ ~ ~ -x Ti 4 and that the analogous relations are imposed on the boundaries of the g~ven domain. Introduction Einstein 's gravitat ional equations 1 G ~ = -~ T ~ (1) imply the four relations T, ~ It,~ = O . (2) Conversely, LICH~C~ROWICZ [1] (see also [2, 3]) has proved that , in the initial value problem, Eq. (1) m ay be replaced by (2) and by the system R~t~= ~ , ~ ( T ~ I T g ~ ) (3) provided t h a t the initial da ta verify the consistency conditions Gi 4 = -~ Ti 4. (4) The importance of this result lies in the fact t h a t it singles out Eq. (2) as a subsystem of Eq. (1). Now, since Eq. (2) have a definite physical meaning by themselves, the above result allows us to s tudy their role in the determinat ion of the solution of the gravi tat ional problem. For the same reason, an analogous possibility is also desirable in more general cases, e.g. in the mixed initial and boundary value problem. I n fact, as we shall prove in a later paper, the analysis of Eq. (2) in a spat ially finite domain of space-time (e.g. in a four-dimensional world1 Latin indices run from 1 to 4. Greek indices run from 1 to 3. The metric tensor is assumed to have the signature ~ 2. A comma indicates partial derivative; a double stroke [like in Eq. (2)] indicates covariant derivative. Einstein's Equations 247 tube) plays an important role in the study of the inertial phenomena. Therefore, the purpose of the presen~ work is to extend LICm~EROW~CZ'S result to the mixed initial and boundary value problem. In § 1 we shall briefly indicate the hypothesis required, as well as the exact formulation of the problem. In § 2 we shall prove the desired result: Eq. (1) may be replaced by the system (2), (3) also in the mixed initial and boundary value problem, provided that the initial data verify the consistency conditions (4), and that the analogous relations (G,~ + z T,~) N ~ = 0 (5) are imposed on the boundaries of the given domain. [In (5), as usual, N ~ denotes the normal to the boundary itself.] § 1 We recall the following definitions [1 -4 ] : the spacetime continuum Va is a twice continuously differentiable manifold on which there is defined a metric d s 2 = g~,~ d x ~ d x ~ of normal hyperbolic type everywhere with signature (÷ ÷ ÷ ) . A coordinate system is physically acceptable if one of its variables is time like, and the other three space-like. On then has g44 < 0 and the reciprocal quadratic forms g4~ g4t~ g~ and g~ g~4 are positive definite. A metric is said regular if it is continuously differentiable (C1, C a piecewise). Since the manifold g~ admits of an everywhere hyperbolic metric, it possesses a vector field oriented in time, and therefore a global system of time lines. In V4 let the domain t2 verify the properties : i) Y2 is the topological product of a manifold Va and the real straight line R (the mappings of R in V4 oriented in time, mappings of V8 everywhere oriented in space) ; ii) the spatial sections, mappings of Va, are bounded manifolds, homcomorphic to spatially finite regions of the Euclidean space Ra. In Q we assume a sufficiently smooth distribution of matter, described by a non-zero stress tensor T ~ . Then we have the Einstein field equations G~.~ = R~r~ 2 R g ~ , n = 7¢ T ~ . ~ , (1)" @default.
• W3148758246 created "2021-04-13" @default.
• W3148758246 creator A5090808958 @default.
• W3148758246 date "2005-01-01" @default.
• W3148758246 modified "2023-09-27" @default.
• W3148758246 title "On the Mathematical Structure of Einstein's Equation, s in Mixed Initial and Boundary Value Problems" @default.
• W3148758246 cites W1680579736 @default.
• W3148758246 cites W3016640939 @default.
• W3148758246 hasPublicationYear "2005" @default.
• W3148758246 type Work @default.
• W3148758246 sameAs 3148758246 @default.
• W3148758246 citedByCount "0" @default.
• W3148758246 crossrefType "journal-article" @default.
• W3148758246 hasAuthorship W3148758246A5090808958 @default.
• W3148758246 hasConcept C114614502 @default.
• W3148758246 hasConcept C118615104 @default.
• W3148758246 hasConcept C121332964 @default.
• W3148758246 hasConcept C134306372 @default.
• W3148758246 hasConcept C146846114 @default.
• W3148758246 hasConcept C182310444 @default.
• W3148758246 hasConcept C26955809 @default.
• W3148758246 hasConcept C2776436953 @default.
• W3148758246 hasConcept C33923547 @default.
• W3148758246 hasConcept C36503486 @default.
• W3148758246 hasConcept C37914503 @default.
• W3148758246 hasConceptScore W3148758246C114614502 @default.
• W3148758246 hasConceptScore W3148758246C118615104 @default.
• W3148758246 hasConceptScore W3148758246C121332964 @default.
• W3148758246 hasConceptScore W3148758246C134306372 @default.
• W3148758246 hasConceptScore W3148758246C146846114 @default.
• W3148758246 hasConceptScore W3148758246C182310444 @default.
• W3148758246 hasConceptScore W3148758246C26955809 @default.
• W3148758246 hasConceptScore W3148758246C2776436953 @default.
• W3148758246 hasConceptScore W3148758246C33923547 @default.
• W3148758246 hasConceptScore W3148758246C36503486 @default.
• W3148758246 hasConceptScore W3148758246C37914503 @default.
• W3148758246 hasLocation W31487582461 @default.
• W3148758246 hasOpenAccess W3148758246 @default.
• W3148758246 hasPrimaryLocation W31487582461 @default.
• W3148758246 hasRelatedWork W113400240 @default.
• W3148758246 hasRelatedWork W1928960068 @default.
• W3148758246 hasRelatedWork W1987431962 @default.
• W3148758246 hasRelatedWork W2005254674 @default.
• W3148758246 hasRelatedWork W2009756463 @default.
• W3148758246 hasRelatedWork W2010761729 @default.
• W3148758246 hasRelatedWork W2039148738 @default.
• W3148758246 hasRelatedWork W2049316036 @default.
• W3148758246 hasRelatedWork W2087083097 @default.
• W3148758246 hasRelatedWork W2090793328 @default.
• W3148758246 hasRelatedWork W2125355722 @default.
• W3148758246 hasRelatedWork W2204376908 @default.
• W3148758246 hasRelatedWork W2327384619 @default.
• W3148758246 hasRelatedWork W2327447718 @default.
• W3148758246 hasRelatedWork W2351217015 @default.
• W3148758246 hasRelatedWork W2488273306 @default.
• W3148758246 hasRelatedWork W2952671711 @default.
• W3148758246 hasRelatedWork W2999037010 @default.
• W3148758246 hasRelatedWork W848127 @default.
• W3148758246 hasRelatedWork W2523721990 @default.
• W3148758246 isParatext "false" @default.
• W3148758246 isRetracted "false" @default.
• W3148758246 magId "3148758246" @default.
• W3148758246 workType "article" @default.