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On the Dynamic Effect of Infinite Redshift Surface: Lorentz Factor Analysis Method Jiayu NI1, 2*, Yuan Tu3 1 The College of Engineering and Technology of CDUT, Sichuan Leshan, 614000, China 2 Center for Theoretical Physics, Sichuan University, Chengdu, 610064, China 3 College of Physics, Sichuan University, Chengdu, 610064, China Abstract The generation of infinite redshift surface is due to the singularity of Lorentz transformation, according to this fact, the dynamic effect of the infinite redshift surface can be analyzed by the singularity of the parameters in the Lorentz transform factor. In this paper, the Lorentz factor analysis method is proposed, and the corresponding interpretation is given in combination with the background of mathematics and physics. And this method is applied to the analysis of Reissner-Nordstrom and Kerr-Newman black hole spacetime, Find that the observation of infinite redshift surface will expand and contract with the observer's radial motion, and the corresponding formula is given quantitatively. We give a new proof of the inaccessibility of the time-like singularity of a charged black hole and prove that the property is only related to the electrification and has nothing to do with the rotation. It is shown that the observation result of the space-like quasi-circular motion observer is consistent with that of the static observer at infinity. The concept of minimum blueshift radius is proposed in the study of the observer who escapes at the speed of light in normal spacetime, its existence makes a stronger constraint on black hole parameters than traditional condition, this is probably the reason why it is difficult to form an intermediate-mass black hole, at the end of the paper, some discussions are made. Keywords: dynamic effect, infinite redshift surface, black hole, time-like singularity chinaXiv:202104.00128v1 1. Introduction At present, the research on event horizon is rich, but the research on infinite redshift surface is not so much[1-4]. Although the horizon has many thermodynamic and quantum mechanical properties[5-6], the infinite redshift surface is of great significance in practical observation, because it is closely related to redshift and blueshift. The Lorentz transformation is an important foundation in general relativity, it gives the transformation relation of physical law between different reference systems. But this transformation is parameter dependent. For example, in Minkowski spacetime of special relativity, this parameter is expressed as relative velocity, and this leads to the singularity of Lorentz transformation at some parameter points. But it doesn't represent the real singularity of spacetime, the scalar formed by the Riemannian curvature tensor at the singular point is probably not divergent[7]. This shows that the singularity of Lorentz transformation corresponds to the infinite redshift surface, which is caused by the singularity of coordinates. From this point of view, we can study the variation of the singularity of Lorentz transform with parameters in various backgrounds of spacetime. We first discuss Reissner-Nordstrom black hole spacetime as the most general spherically symmetric spacetime, because of the symmetry requirement, we only need to discuss the radial case. Secondly, the Kerr-Newman black hole space-time is studied as the most general spacetime in general relativity, at this time, we need to study not only the radial case but also the observation effect caused by the angular movement. Besides, we are also interested in another problem, that is, in normal spacetime, what is the infinite redshift surface when escaping at the speed of light? 1 This paper studies these problems and related problems, and adopts metric sign convention is , and uses units with . 2. Lorentz factor analysis method The metric of Minkowski spacetime is simple and very basic, the analysis method can be made clear by starting from this. Specifically, we need to change the metric parameter to the time parameter, such as replace with and with , where is the radial velocity and is the angular velocity. In this way, the singularity of the metric can be expressed uniformly in the form of a time parameter, as we did in the Lorentz transformation. Consider a flat metric (1) the significance of the second equation is to extract the velocity parameters. Equation (1) is the observation result of the static observer, compare it with the result of the follow-up observer ( , is the proper time), and we can get the Lorentz transform of the time (2) We are used to using a symbol to represent the factor of Lorentz transformation. In the example above, . The conditions of Lorentz transformation include the existence of factor, this means that the root in the factor must be meaningful, the factor itself should be meaningful too. We can use this to analyze the infinite redshift surface of black hole spacetime. The phenomenon of infinite redshift can be attributed to the fact that the proper time is zero on a hypersurface in spacetime, from the perspective of coordinate transformation between static observer and follow up observer, it is equivalent to the Lorentz factor at a singular point. If we take the parameter as the velocity on the hypersurface of spacetime manifold, this will give that the speed cannot be the speed of light , otherwise, the Lorentz transformation between the static observer and the follow-up observer is singular. This shows that the essence of the infinite redshift phenomenon is the singularity of Lorentz transformation and that the infinite redshift surface is indeed a null hypersurface. For the velocity on hypersurface, we should consider it as the derivative of space-like coordinates to time-like coordinates. In this way, in normal spacetime, , the world line of an object with mass is time-like, so it has . However, there is the one-way membrane region of spacetime exchange in black hole spacetime, and the definition of velocity on the hypersurface becomes , then the velocity of the object with mass will be in a range . It should be noted that in the unidirectional membrane area, the future direction of the light cone is single, and all physical processes are unidirectional, so the velocity parameter on the hypersurface exceeds the speed of light and does not violate any laws of physics[8]. chinaXiv:202104.00128v1 After defining these basic concepts, we will use these concepts to analyze the dynamic effect of infinite redshift surface through the singularity of Lorentz transformation. 3. General spherically symmetric spacetime According to the Birkhoff theorem, in general relativity, the most general spherically symmetric black hole is Reissner- Nordstrom(R-N) black hole[9], we will take this spacetime as the research object of this section. Since it is a spherically symmetric spacetime, we are not interested in the motion around the black hole, so we only study the radial motion. Write directly the spacetime metric with velocity parameter (3) Where is a two-dimensional spherical hypersurface. is the radial coordinate, and are mass and charge, respectively. We find the Lorentz factor in this spacetime 2 (4) Let's analyze the singularity of equation (4), because we want to study the dynamic effect of infinite redshift surface, we need to exclude the infinite point first. The zero points of the molecule, , is the inner and outer infinite redshift surface of the traditional R-N black hole (5) The results show that the infinite redshift surface coincides with the event horizon in static spacetime, this is related to the symmetry of spacetime, so equation (5) is also the event horizon of the R-N black hole. The denominator has four zero points, which we denote as (6) For convenience, we express them uniformly as . When , the singularity of equation (4) only depends on , this is the traditional situation. When , the zero points of the denominator are , not , equation (4) is singular at , the two-dimensional spherical hypersurface at , , becomes infinite redshift surface. But we need to notice that and should be eliminated from the physical point of view. If they make sense, then is required, this is an unreasonable and very strong condition, we should only require to avoid the problem of negative energy. We know that the infinite redshift surface is a coordinate singularity, and its curvature of spacetime does not diverge. Its influence only exists in the observation. Even so, and give a very rich dynamic effect of the infinite redshift surface. Firstly, the inner infinite redshift surface gives the property that the time-like singularity cannot be approached. There has been a lot of related work on the property that time-like singularity cannot be approached[10-12]. Because the region within the inner infinite redshift surface of the R-N black hole is a normal spacetime, that is, is time-like, and is space-like, the essential singularity of the R-N black hole is a time-like singularity. In normal spacetime, , to make the conclusion more significant, we take the limit case (7) chinaXiv:202104.00128v1 This is a very important result, it seems that the inner infinite redshift surface will contract inward as the observer moves, as long as the black hole is charged, , the inner infinite redshift surface will never shrink to the essential singularity. But if , the inner infinite redshift surface will shrink to the essential singularity, which is what happens in the Schwarzschild black hole. This result shows that it is impossible to approach the essential singularity in a charged black hole. The change rate of the inner infinite redshift surface with velocity is given here (8) When and are known, this rate is negative and increases with the increase of . Then, we should consider the expansion of the outer infinite redshift surface . The change rate of with velocity is given 3 (9) It can be proved numerically that with the increase of , the rate given by equation (8) is very small, while the rate given by equation (9) is surprisingly large, especially when is close to 1.
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