Vacuum solution with nonzero cosmological constant

In summary, the equation for the radial geodesic in the vacuum solution to Einstein's equations with non-zero cosmological constant takes the form E = T + V, but with a term proportional to r^2 in V(r). This does not match with ordinary Newtonian gravity and raises the question of why a non-zero cosmological constant is believed to exist. However, the tiny magnitude of the constant makes it undetectable on smaller scales, and recent astronomical evidence suggests its existence. The condition for it to be consistent with Newtonian gravity is that Lambda R^2 would have to be much less than 1, where R is the maximum distance over which Newtonian gravity has been observed to hold. Current estimates of the cosmological constant
  • #1
La Guinee
24
0
Consider the vacuum solution to Einstein's equations with non-zero cosmological constant. Following Carroll, we can find the equation for the radial geodesic with the aid of killing vectors. It takes the standard form: E = T + V. But, with non-zero cosmological constant V(r) now has a term proportional to r^2. This obviously doesn't reduce to ordinary Newtonian gravity. If there were actually a term proportional to r^2 we would have experimentally detected it. What's going on?
 
Physics news on Phys.org
  • #2
La Guinee said:
Consider the vacuum solution to Einstein's equations with non-zero cosmological constant. Following Carroll, we can find the equation for the radial geodesic with the aid of killing vectors. It takes the standard form: E = T + V. But, with non-zero cosmological constant V(r) now has a term proportional to r^2. This obviously doesn't reduce to ordinary Newtonian gravity. If there were actually a term proportional to r^2 we would have experimentally detected it. What's going on?
General Relativity with a non zero cosmological constant is not compatible with special relativity in flat spacetimes or Newtonian gravity in the limit.
 
  • #3
MeJennifer said:
General Relativity with a non zero cosmological constant is not compatible with special relativity in flat spacetimes or Newtonian gravity in the limit.

Ok. But if a nonzero cosmological constant has an r^2 potential it can't be compatible with our universe either. How does one reconcile this with the fact that people think the cosmological constant isn't zero?
 
  • #4
La Guinee said:
But if a nonzero cosmological constant has an r^2 potential it can't be compatible with our universe either.
How do you conclude as such?
 
  • #5
MeJennifer said:
How do you conclude as such?

Wouldn't we have experimentally detected a potential that grows as r^2?
 
  • #6
La Guinee said:
Wouldn't we have experimentally detected a potential that grows as r^2?
Not if the constant of proportionality is exceedingly tiny!

The effect of a tiny cosmological constant is only noticeable on huge (cosmological!) scales, and it's only in the last decade or so that we've found astronomical evidence to suggest it isn't zero.
 
  • #7
DrGreg said:
Not if the constant of proportionality is exceedingly tiny!

The effect of a tiny cosmological constant is only noticeable on huge (cosmological!) scales, and it's only in the last decade or so that we've found astronomical evidence to suggest it isn't zero.

I don't see how that makes sense. No matter how small the cosmological constant is, I can find an r much much greater than the cosmological constant. Unless you're saying the cosmological constant is small even compared to the size of the universe.
 
  • #8
La Guinee said:
I don't see how that makes sense. No matter how small the cosmological constant is, I can find an r much much greater than the cosmological constant. Unless you're saying the cosmological constant is small even compared to the size of the universe.

I take back what I just said. The condition would be that Lambda R^2 would have to be much less than 1, where R is the maximum distance over which Newtonian gravity has been observed to hold. Is this consistent with current estimates of the cosmological constant? In particular, the current upper bound would have to be less than or equal to the upper bound implied by this criterion. Anyone know if this is true?
 
  • #9
I found a paper that gives the estimate Lambda ~ 10^-52. That is indeed pretty small.
 

1. What is meant by vacuum solution with nonzero cosmological constant?

A vacuum solution with nonzero cosmological constant refers to a solution in Einstein's field equations that describes the behavior of spacetime in the presence of a nonzero cosmological constant, also known as dark energy. This solution describes the expansion of the universe and the acceleration of its expansion due to the presence of dark energy.

2. How does a nonzero cosmological constant affect the behavior of spacetime?

A nonzero cosmological constant causes the expansion of the universe to accelerate, as opposed to the deceleration predicted by the standard model of cosmology. It also affects the curvature of spacetime, leading to a positive curvature and a closed universe.

3. What is the significance of a vacuum solution with nonzero cosmological constant?

The existence of a vacuum solution with nonzero cosmological constant has major implications for our understanding of the universe. It explains the observed acceleration of the expansion of the universe and provides evidence for the existence of dark energy, a mysterious force that makes up about 70% of the energy in the universe.

4. How is a vacuum solution with nonzero cosmological constant different from the Lambda-CDM model?

The Lambda-CDM (cosmological constant plus cold dark matter) model is a specific model of the universe that includes both a nonzero cosmological constant and cold dark matter. A vacuum solution with nonzero cosmological constant is a general solution that describes the behavior of spacetime in the presence of a nonzero cosmological constant, regardless of the presence of cold dark matter or other forms of matter.

5. What are some possible implications of a vacuum solution with nonzero cosmological constant for the future of the universe?

One possible implication is that the universe will continue to expand at an accelerating rate, leading to a future in which galaxies will become increasingly isolated from one another. Another implication is that the universe may eventually experience a "Big Rip," in which the expansion becomes so rapid that it tears apart all matter and even spacetime itself.

Similar threads

I EFE with cosmological constant
    • Special and General Relativity
Replies
7
Views
513
I Cosmological Constant Term in Einstein Equations
    • Special and General Relativity
Replies
2
Views
1K
I Sign of vacuum energy density
    • Quantum Physics
Replies
3
Views
416
I Einstein Field Eqns: East/West Coast Metrics
    • Special and General Relativity
Replies
6
Views
1K
I Question about cosmological constant
    • Special and General Relativity
2
Replies
43
Views
4K
B Cosmological constant for the Universe's expansion
    • Cosmology
Replies
5
Views
1K
A Why isn't cosmological constant a mass of the graviton?
    • Special and General Relativity
Replies
21
Views
2K
I Normalization Constant in Einstein-Hilbert Action
    • Special and General Relativity
Replies
10
Views
1K
I Einstein, Friedman and the cosmological constant
    • Cosmology
Replies
5
Views
1K
Cosmological constant term and metric tensor
    • Special and General Relativity
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top