Then it seems the problem is hard to solve as you have said...I am not sure that I canIbix said:
Well, can you name some symmetries? I can think of six in two sets of three. Hint: what is the defining property of FLRW spacetimes?Arman777 said:
Homoegenous and Isotropic spaceIbix said:
NoArman777 said:
No. What the comments from @Ibix and @Orodruin are telling you is that, instead of looking for a mechanical process to crank out Killing vectors from the metric, you should try to understand physically what the symmetries of the spacetime geometry are, and then use that understanding to come up with an ansatz for what you expect the Killing vectors to be.Arman777 said:
Yeah, I can't imagine solving the Killing equation in any other way than guessing a coordinate system where the metric is independent of some coordinate.Ibix said:
That method won't always find all of the Killing vectors, though. For example, for spherical symmetry there is a 3-parameter group of Killing vectors, but looking for an adapted coordinate chart will only find one of them.stevendaryl said:
Well. There’s nothing stating that you cannot use other coordinate systems to find the remaining ones.PeterDonis said:
Yes, that's true, although it does exchange the problem of finding all the Killing vectors for the problem of finding all the coordinate systems.Orodruin said:
Again, true, but I'm not sure how helpful it is if you don't already know what the Killing field is.Orodruin said:
Yes, good point; I believe this is one way to get all three of the spherical symmetry Killing fields, for example.Orodruin said:
Sure, you are still guessing. But for some seeing coordinate systems where symmetry is apparent may be easier than seein the Killing fields.PeterDonis said:Yes, that's true, although it does exchange the problem of finding all the Killing vectors for the problem of finding all the coordinate systems.
If you have two of them, then yes. Taking their connutator will give the third.PeterDonis said:
The FLRW metric is a mathematical model used in cosmology to describe the geometry of the universe. It is based on the assumptions of homogeneity and isotropy, meaning that the universe looks the same in all directions and at all points in time.
Killing vectors are mathematical operators that represent symmetries in a given system. In the context of the FLRW metric, they represent the symmetries of the universe, such as rotational or translational symmetries.
Finding Killing vectors in the FLRW metric allows us to better understand the symmetries and dynamics of the universe. It also helps us to solve equations and make predictions about the behavior of the universe.
The simple equation for finding Killing vectors in the FLRW metric is given by: K^{\mu} = a(t)\partial_t + b(t)\partial_r + c(t)r\partial_{\theta} + c(t)r\sin\theta\partial_{\phi} where a(t), b(t), and c(t) are functions of time.
Some applications of finding Killing vectors in the FLRW metric include studying the evolution and dynamics of the universe, making predictions about the behavior of the universe, and testing different cosmological models.