By Christian Klein
Exact ideas to Einstein’s equations were necessary for the knowledge of common relativity in lots of respects. they've got ended in such actual suggestions as black holes and occasion horizons, and helped to imagine fascinating beneficial properties of the idea. This quantity reviews the options to the Ernst equation linked to Riemann surfaces intimately. moreover, the booklet discusses the actual and mathematical features of this classification analytically in addition to numerically.
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Extra resources for Ernst equation and Riemann surfaces : analytical and numerical methods
This makes the following two expressions identical: ⎧Ψ = R0 RG ( sin ϕ − i cos ϕ ) ⎪ ⎛ ⎞ . ⎨ RG −1⎟ ⎪ ΔR = R0 ⎜⎜ 1 − i ⎟ R0 ⎝ ⎠ ⎩ We also have, therefore, the complex conjugate function Ψ = R0 RG ( sin ϕ + i cos ϕ ) The displacement velocity In the complex framework, we can now develop the expression for the velocity differential Δυ pertinent to the vacuum displacement as acceleration in general, or more particularly in the case of gravitation, from the preceding: Geometry of Acceleration in Space-Time ⎛ R Δυ = c ⋅ ⎜1 − 1 − G ⎜ R0 ⎝ ⎛ R ≡ c ⋅ ⎜⎜1 − i G ⋅ R0 ⎝ ⎞ ⎟⎟ ⎠ R 1− 0 RG ⎞ ⎟⎟ ⎠ 21 .
Our model may be thought of as a starting point for more detailed investigations of the non-baryonic dark matter scenario. As we see here, the dilaton (massless scalar field) may act dynamically in the same way as massive bodies. In this way the dilaton can become another candidate for the cold dark matter. On the galactic scale, however, where dilatonic configurations could be invoked to explain the central engine for AGNs and quasars, our model does not solve the flat rotation curve problem. In this aspect it would be very interesting to extend the model discussed in this paper by inclusion of an axion field.
It contains 16 independent components. The remaining 4 functions allowed by the general form of the material tensor density η μνλρ ( x ) are provided by the 4 independent components of the totally antisymmetric G tensor jμνλ ( x ) formed from ρ M ( x ) and jM ( x ) . It is erroneous to try to relate this antisymmetric tensor to the external current, as in (21), because this will introduce an additional 16 new functions. It is also erroneous to treat this tensor as any kind of current, because it is not a tensor density.
Ernst equation and Riemann surfaces : analytical and numerical methods by Christian Klein