instead of null infinity defined in terms of asymptotically flat regions that null geodesics go to you instead define it in terms of asymtptotically AdS regions
His thesis was "Observation of Einstein-Podolsky-Rosen Entanglement on Supraquantum Structures by Induction Through Nonlinear Transuranic Crystal of Extremely Long Wavelength (ELW) Pulse from Mode-Locked Source Array"
@0celo7 Can you visualize $X'' + X = 0$ as saying that the acceleration vector is just a scaled multiple of the position vector, $X'' = - X$, and that this must be true for all values of the parameter so it's obvious the varying parameter is just the angle at which the vector is oriented in the plane,
thus a solution could either be an anti-clockwise-rotating circle of radius $A$ or an clockwise-rotating circle of radius $B$, giving as general solution a supposition of these generating an ellipse $X(t) = Ae^{it} + Be^{-it}$? Constants like $\lambda$ then tell you how fast the circles rotate, and you can see why they are indexed etc... and boundary conditions are about orienting the circles. Visualied here from Needham's book P242: https://i.sstatic.net/Jjj62.png
@Slereah Let $\{E_i\}$ be an orthonormal frame. Then $\mu$ is the canonical volume form iff $\mu(E_1,\dotsc, E_n)=1$. Then $$(\nabla_X\mu)(E_1,\dotsc,E_n)=\nabla_X(\mu(E_1,\dotsc,E_n))-\sum \mu(E_1,\dotsc,\nabla_X E_i,\dotsc,E_n)=-\sum g(E_i,\nabla_X E_i)\mu(E_1,\dotsc,E_i,\dotsc,E_n)=-\sum g(E_i,\nabla_X E_i)=-\frac{1}{2}\sum \nabla_X g(E_i,E_i)=0$$
there might be some plusses and minuses in the Lorentzian case
but it works essentially the same
@Slereah Apparently one can define an adjoint of $\nabla$ that's useful for [things]
I'm trying to understand how the expansion of the universe affects space-time
it should... accelerate it? (Although I read time is slowing, but it might be to counterpart the energy needed for the expansion of the universe, not related to my question)
As more dense matter slows down time, less dense... would make it faster? Compared to current expansion.
If $V$ is a finite dimensional Hilbert space for any vector $x \in V$ and endomorphism $A$, the function
$$
Q(x) = \langle Ax, x \rangle
$$
defines a quadratic form on $V$. Now, I would like to show that $Q$ is continuous. The main thought I have about this is that it is very similar to the (an...
If $V$ is a finite dimensional Hilbert space for any vector $x \in V$ and endomorphism $A$, the function
$$
Q(x) = \langle Ax, x \rangle
$$
defines a quadratic form on $V$. Now, I would like to show that $Q$ is continuous. The main thought I have about this is that it is very similar to the (an...
