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Constant Gaussian curvature, FLRW conditions and a universe without matter
International Journal of Geometric Methods in Modern Physics ( IF 1.8 ) Pub Date : 2022-07-06 , DOI: 10.1142/s0219887822501717
Wladimir-Georges Boskoff 1
Affiliation  

The problem of FLRW universes can be seen in the excellent works of A. Friedmann, G. Lemaître, H. P. Robertson, A. G. Walker. The key of obtaining FLRW universes with or without cosmological constant is related to the special form of the contravariant energy–momentum tensor which describes a perfect fluid with components Tij=(ρ0+p0)uiujp0gij because such a “matter” leads to FLRW conditions R11=R22=R33 which allow to obtain a texture compatible with Hubble law. The results obtained in the works presented as references show very complete analysis about how the constant curvature 2D metric dΩ2=d𝜃2+sin2𝜃dϕ2 can create possible universes and their evolution. Can we expect to have some other FLRW universes if we use a 2D spatial part as dΩA2=d𝜃2+A2(𝜃)dϕ2?

The answer is yes and we prove the special role of the constant ratio K=A(𝜃)A(𝜃) in the creation of FLRW universes. The solutions obtained lead to FLRW universes which come from surfaces having constant Gaussian curvature related by the previous K from the rule KG=K. Therefore, in the case of the 2D metrics dΩA2=d𝜃2+A2(𝜃)dϕ2, only the ones having constant Gaussian curvature create FLRW universes. In fact, we find all possible FLWR-type solutions of Einstein field equations coming from constant Gaussian curvature geometries. We complete our study about possible FLRW universes with an example of a spacetime that cannot be “filled” with matter using the above FLRW conditions. This becomes a new example of universe without matter satisfying the Einstein’s field equation with cosmological constant.



中文翻译:

恒定高斯曲率、FLRW 条件和无物质宇宙

FLRW 宇宙的问题可以在 A. Friedmann、G. Lemaître、HP Robertson、AG Walker 的优秀作品中看到。获得有或没有宇宙学常数的 FLRW 宇宙的关键与描述具有分量的完美流体的逆变能量-动量张量的特殊形式有关一世j=(ρ0+p0)一世j-p0G一世j因为这样的“事情”会导致 FLRW 条件R11=R22=R33这允许获得与哈勃定律兼容的纹理。在作为参考的作品中获得的结果显示了关于恒定曲率 2D 度量如何的非常完整的分析dΩ2=d𝜃2+2𝜃dφ2可以创造可能的宇宙及其演化。如果我们使用 2D 空间部分作为dΩ一个2=d𝜃2+一个2(𝜃)dφ2?

答案是肯定的,我们证明了恒比的特殊作用ķ=一个(𝜃)一个(𝜃)在创造 FLRW 宇宙中。获得的解决方案导致 FLRW 宇宙来自具有与前面相关的恒定高斯曲率的表面ķ从规则ķG=-ķ. 因此,在二维度量的情况下dΩ一个2=d𝜃2+一个2(𝜃)dφ2,只有具有恒定高斯曲率的那些才能创建 FLRW 宇宙。事实上,我们发现爱因斯坦场方程的所有可能的 FLWR 型解都来自恒定高斯曲率几何。我们通过使用上述 FLRW 条件无法“充满”物质的时空示例来完成对可能 FLRW 宇宙的研究。这成为没有物质满足爱因斯坦场方程的宇宙学常数的新例子。

更新日期:2022-07-06
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