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exponent137
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Why cosmological constant term ##\Lambda g_{uv}## in Einstein equation is proportional to ##g_{uv}##. Why it is even proportional to ##g_{uv}## in spacetime of MInkowski?
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I have a detailed discussion of this in section 8.1.4 of my GR book, http://www.lightandmatter.com/genrel/ , but a quick and dirty argument is that if we have in mind something that's a form of stress-energy built into the structure of space itself, then its stress-energy tensor acts like a special tensor that's built into the structure of spacetime. One way of stating the equivalence principle is that the only such built-in tensor is the metric itself.exponent137 said:Why cosmological constant term ##\Lambda g_{uv}## in Einstein equation is proportional to ##g_{uv}##.
Minkowski space isn't a solution of the Einstein field equations with a nonzero cosmological constant.exponent137 said:Why it is even proportional to ##g_{uv}## in spacetime of MInkowski?
In his paper Inside Gravity* Prof Padmanabhan refers to the fact that the Einstein-Hilbert action can be split in a bulk part and a surface part. If one discards the bulk part and extremizes the action then the EFE come out unchanged and the cosmological constant emerges as a constant of integration. So it can be put in or appear naturally with much the same result.exponent137 said:Why cosmological constant term ##\Lambda g_{uv}## in Einstein equation is proportional to ##g_{uv}##. Why it is even proportional to ##g_{uv}## in spacetime of MInkowski?
Mentz114 said:More importantly, if any metric is multiplied by a constant this is the same as putting a constant curvature everywhere. ( As ben says, I think)
Sorry. My misunderstanding. I was thinking of de Sitter space where the solution of the EFE is ##G_{\mu\nu}=-\Lambda g_{\mu\nu}##. This spacetime has no matter but has constant curvature everywhere ( that is how I interpret "four dimensional spaces of constant and isotropic curvature").bcrowell said:Huh? I don't understand what you mean by this.
The cosmological constant term, denoted by Λ (Lambda), is a mathematical constant that was originally introduced by Albert Einstein in his theory of general relativity. It represents the energy density of the vacuum of space and is responsible for the acceleration of the expansion of the universe. It is important in cosmology because it helps explain the observed accelerated expansion of the universe and is also believed to play a role in the formation and evolution of large-scale structures in the universe.
In general relativity, the metric tensor is a mathematical object that describes the curvature of spacetime. The cosmological constant term is included in the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy. The presence of the cosmological constant term in these equations affects the curvature of spacetime and therefore, the metric tensor.
The cosmological constant term is currently believed to be a true constant, meaning it does not vary over time. However, there have been theories and hypotheses that suggest it may not be a constant and could vary over time. This is an active area of research in cosmology and there is ongoing debate and investigation on this topic.
The inclusion of the cosmological constant term in the Einstein field equations affects the solutions to these equations in several ways. It can change the overall geometry of spacetime, leading to different predictions for the behavior of the universe. It also affects the rate of expansion of the universe, as well as the formation and evolution of structures within the universe.
Currently, there is no known physical interpretation of the value of the cosmological constant term. It is a free parameter in the equations of general relativity and its value is not predicted by any known theory. This has led to the term "cosmological constant problem", as the value of Λ is significantly smaller than what would be expected based on theoretical predictions. This is an active area of research in cosmology and there are ongoing efforts to better understand the physical significance of the cosmological constant term.