3D-BRANE GRAVITY WITHOUT EQUIDISTANCE POSTULATE. Ruslan Sharipov Abstract. The 3D-brane universe model is an alternative non-Einsteinian theory of gravity. It was initially built using the so-called equidistance postulate. In this paper we consider a version of the theory without this equidistance postulate. 1. Introduction. The 3D-brane universe model is based on the idea that the four-dimensional spacetime subdivides into a dense foliation of 3D-branes. Each brane represents some definite instantaneous state of the real 3D-universe in its evolution pathway as shown in Fig. 1.1. Arguments in favor of the 3D-brane universe model opposed to the standard 4D paradigm are given in [1] (see also [2] and [3]). We can choose the normal vector n of the unit length directed to the future at each point of each 3D-brane. Since 3D- branes are dense in the four-dimensional spacetime, these vectors n constitute a vector field. Integral curves (see § 2.12 of Chapter 2 in [4]) of the vector field n are shown as black lines in Fig. 1.1. The arrows at the ends of these curves indicate the direction of the time evolution from the past to the future. Let’s choose some 3D-brane for the initial one, say it is the brane AA’ in Fig. 1.1. Let’s choose some coordinates x 1 ,x 2 ,x 3 in the initial 3D-brane. We can extend them to other 3D-branes along the integral curves of the field n by setting x i (B)= x i (A), i =1, 2, 3, (1.1) for any point A of the initial 3D-brane. The coordinates introduced in such a way are called comoving coordinates (see [5]). An observer whose world line (see [6]) coincides with one of the integral curves of the vector field n in Fig. 1.1 is called a comoving observer. The lengths of the curves AB and A ′ B ′ can be used for introducing time intervals: t = |AB| c gr , t ′ = |A ′ B ′ | c gr . (1.2) 2010 Mathematics Subject Classification. 53B50, 53Z05, 83C05, 83F05. Key words and phrases. 3D brane universe, gravity equations. Typeset by A M S-T E X