I Spatially Homogeneous Scalar Field on Spacetime: Showing $\nabla^2 f$

  • I
  • Thread starter Thread starter ergospherical
  • Start date Start date
  • Tags Tags
    Mean
ergospherical
Science Advisor
Homework Helper
Education Advisor
Insights Author
Messages
1,097
Reaction score
1,384
If ##f## is a "spatially homogeneous" scalar field on spacetime ##ds^2 = dt^2 - a^2(t) \delta_{ij} dx^i dx^j## then show that ##\nabla^2 f = \ddot{f} + 3H \dot{f}##. Should be easy if I knew what the condition on ##f## is, i.e. ##\nabla^2 f = \partial_{\mu} \partial^{\mu} f = \ddot{f}- a^{-2}(t) \delta^{ij} \partial_i \partial_j f = \dots##?
 
Physics news on Phys.org
Spatially homogeneous = does not depend on spatial position. In other words, ##f## is a function of ##t## only.
 
  • Like
Likes FactChecker, vanhees71 and ergospherical
I messed up the double covariant, should be:\begin{align*}
\nabla^2 f &= \nabla_{\mu}(\partial^{\mu} f) \\
&= \partial_{\mu} \partial^{\mu} f + g^{\nu \rho} \Gamma_{\mu \nu}^{\mu} \partial_{\rho} f \\
&= \partial_t^2 f + 3H \partial_t f
\end{align*}(##\Gamma_{0i}^{j} = H\delta^{j}_{i}##)
 
Indeed. Only the time derivatives survive since the field only depends on t. Then it is just a matter of finding the appropriate trace of Christoffel symbols.
 
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...
I started reading a National Geographic article related to the Big Bang. It starts these statements: Gazing up at the stars at night, it’s easy to imagine that space goes on forever. But cosmologists know that the universe actually has limits. First, their best models indicate that space and time had a beginning, a subatomic point called a singularity. This point of intense heat and density rapidly ballooned outward. My first reaction was that this is a layman's approximation to...
Back
Top