Einstein Equation with Cosmological Constant & Variable x^u

In summary, the action \int (k(R-2\Lambda)+L_m)\sqrt{g} leads to the Einstein equation R_{uv}-\frac{1}{2}R g_{uv}-\Lambda g_{uv}=k'T_{uv}. This remains the same even if \Lambda is a function of the coordinates x^u. However, there is a problem with energy conservation if \Lambda is not a constant, as it affects the covariant derivative in the equation.
  • #1
alejandrito29
150
0
with the action [tex] \int (k(R- 2 \Lambda)+L_m) \sqrt{g} [/tex] the Einstein equation is:

[tex]R_{uv}-\frac{1}{2}R g_{uv}- \Lambda g_{uv} = k' T_{uv}[/tex]

How is the Einstein equation if [tex]\Lambda=\Lambda(x^u)[/tex]? with [tex]x^u[/tex] a coordinate
 
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  • #2
The same. You derive the Einstein equations with a variation with respect to the metric, not the coordinates. So you obtain the Einstein equations

[tex]
R_{uv}-\frac{1}{2}R g_{uv}- \Lambda(x) g_{uv} = ksingle-quote T_{uv}
[/tex]

However, you're in serieus trouble with energy conservation. A constant CC is allowed, because then

[tex]
\nabla_{\rho}(\Lambda g_{\mu\nu}) = \Lambda \nabla_{\rho}g_{\mu\nu}=0
[/tex]

If Lambda is general, energy conservation is spoiled.
 

1. What is the Einstein Equation with Cosmological Constant & Variable x^u?

The Einstein Equation with Cosmological Constant & Variable x^u is a mathematical equation developed by Albert Einstein to describe the relationship between the curvature of space-time and the distribution of matter and energy within it. It includes a cosmological constant, which represents the energy density of the vacuum, and a variable x^u, which accounts for the presence of additional energy sources.

2. How does the Einstein Equation with Cosmological Constant & Variable x^u relate to the theory of relativity?

The Einstein Equation is a fundamental part of Einstein's theory of general relativity, which explains how gravity works on a large scale. It is derived from the principle of equivalence, which states that the effects of gravity are equivalent to the effects of acceleration. The equation allows us to calculate the curvature of space-time caused by the presence of matter and energy.

3. What is the significance of the cosmological constant in the Einstein Equation?

The cosmological constant, represented by the Greek letter lambda (Λ), was originally introduced by Einstein to account for the static nature of the universe. However, after the discovery of the expansion of the universe, the cosmological constant has been used to explain the observed acceleration of the expansion. It represents the energy density of the vacuum and has a crucial role in our understanding of the universe's evolution.

4. How is the variable x^u determined in the Einstein Equation?

The variable x^u represents the presence of additional energy sources, such as dark energy or exotic matter. It can be determined through observations and measurements of the universe's expansion and the distribution of matter within it. However, the exact nature and origin of these additional energy sources are still not fully understood, and further research is needed to determine their exact contribution to the Einstein Equation.

5. Can the Einstein Equation with Cosmological Constant & Variable x^u be solved exactly?

No, the Einstein Equation is a set of complex differential equations that cannot be solved exactly. However, using various techniques and approximations, scientists are able to obtain solutions that accurately predict the behavior of the universe on a large scale. These solutions have been confirmed through numerous observations and experiments and have greatly contributed to our understanding of the universe.

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