Exploring Cosmological Constants

In summary: Assuming my guess is correct, I think the answer is no. The construction proposed reduces to ##\lambda_{\mu\nu}{}^{a_1}{}_{a_1}{}^{a_2}{}_{a_2}...{}^{a_n}{}_{a_n}=\lambda_{\mu\nu}##, a rank two tensor, which you are only free to add if its covariant derivative is zero.
  • #1
jk22
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Can we explain the cosmological constant in this way ?

The Einstein tensor is derived from the Ricci tensor and one property is that, like the stress-energy tensor, its covariant derivative shall vanish.

Since the covariant derivative of the metric vanishes it can be added to the EFE as ##+\Lambda g_{\mu\nu}##

But then we could add any power of the metric like ##+\Lambda_1 g_{\mu\rho}g^\rho_\nu## ?
 
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  • #2
jk22 said:
But then we could add any power of the metric like ##+\Lambda_1 g_{\mu\rho}g^\rho_\nu## ?
Contract over ##\rho## and this becomes ##\Lambda_1 g_{\mu\nu}##. This is true for any product of repetitions of the metric that only has two free indices.
 
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  • #3
jk22 said:
Since the covariant derivative of the metric vanishes it can be added to the EFE

Yes, this is actually the more or less standard argument among physicists for why the cosmological constant term should be expected to be present.
 
  • #4
Does ##(g_{\mu\nu})^2## also give the metric ?
 
  • #5
jk22 said:
Does ##(g_{\mu\nu})^2## also give the metric ?

What do you mean by ##(g_{\mu\nu})^2##?
 
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  • #6
jk22 said:
Does ##(g_{\mu\nu})^2## also give the metric ?
Interpreting ##\left(g_{\mu\nu}\right)^2## literally as ##g_{\mu\nu}g_{\mu\nu}## it's an illegal expression. Interpreting it as ##g_{\mu\nu}g^{\mu\nu}## it's ##\delta_\mu^\mu=4##.
 
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  • #7
Ibix said:
Interpreting ##\left(g_{\mu\nu}\right)^2## literally as ##g_{\mu\nu}g_{\mu\nu}## it's an illegal expression. Interpreting it as ##g_{\mu\nu}g^{\mu\nu}## it's ##\delta_\mu^\mu=4##.
A similar problem arises when you naively would construct a mass term for the graviton in spin-2 field theories, as compared to spin-0 and spin-1.
 
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  • #8
haushofer said:
A similar problem arises when you naively would construct a mass term for the graviton in spin-2 field theories, as compared to spin-0 and spin-1.
Learning QFT is still on my to-do list. One of these days... :oldfrown:
 
  • #9
I mean a special case of ##\lambda_{\mu\nu}^{a_1b_1a_2b_2...a_nb_n}g_{a_1b_1}g_{a_2b_2}...g_{a_nb_n}##
 
  • #10
jk22 said:
Does ##(g_{\mu\nu})^2## also give the metric ?
PeterDonis said:
What do you mean by ##(g_{\mu\nu})^2##?
jk22 said:
I mean a special case of ##\lambda_{\mu\nu}^{a_1b_1a_2b_2...a_nb_n}g_{a_1b_1}g_{a_2b_2}...g_{a_nb_n}##
So, do you mean ##\lambda_{\mu\nu}^{ab}g_{ab}##? What is ##\lambda##?
 
  • #11
##\lambda## were a 2+2n constant tensor. It lacks a sum over n to be complete.
 
  • #12
jk22 said:
##\lambda## were a 2+2n constant tensor.

What 2 + 2n constant tensor? What are you talking about?
 
  • #13
I think OP is proposing ##\lambda## as a tensorial cosmological constant (or at least, asking if such a thing is possible). As I think I've said to you before, @jk22, you must always define your terms. I think I've guessed your meaning, but it is a guess and guesswork is no basis for scientific communication.

Assuming my guess is correct, I think the answer is no. The construction proposed reduces to ##\lambda_{\mu\nu}{}^{a_1}{}_{a_1}{}^{a_2}{}_{a_2}...{}^{a_n}{}_{a_n}=\lambda_{\mu\nu}##, a rank two tensor, which you are only free to add if its covariant derivative is zero. So this is either an over-complicated way of writing ##\Lambda g_{\mu\nu}## or it's not allowed.
 

FAQ: Exploring Cosmological Constants

1. What are cosmological constants?

Cosmological constants are numerical values that are used in equations to describe the behavior and properties of the universe. They are constants because they do not change over time or space.

2. How do cosmological constants affect the universe?

Cosmological constants play a crucial role in determining the expansion rate and structure of the universe. They also affect the amount of matter and energy in the universe, as well as the overall curvature of space.

3. What is the significance of exploring cosmological constants?

Exploring cosmological constants allows us to better understand the fundamental laws of the universe and how it has evolved over time. It also helps us to make predictions about the future of the universe and its ultimate fate.

4. How do scientists measure and study cosmological constants?

Scientists use a variety of tools and techniques, such as telescopes, satellites, and computer simulations, to study cosmological constants. They also analyze data from cosmic microwave background radiation and other astronomical observations.

5. Can cosmological constants change over time?

According to current theories, cosmological constants are considered to be constant values. However, there are some theories that suggest they may have varied in the past or could potentially change in the future. Further research and observations are needed to fully understand this aspect of the universe.

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