Well here we are after all these years, finally isolate concretely why science forbid back to the future style time travel to be a possible reality of time travel (This will surely change my view of time travel stories in general)
What prevents the field to be a function of both spacetime and proper time, is it because it is lorentz invarient thus it must have a unique value in spacetime that agrees to all observers?
I am not sure, probably a half baked understanding of what Slereah previously shared to me about something branched something spacetimes or something, I don't remember
@Slereah (back to ordinary questions). The general approach in solving them is to compute the curvature tensor and then work out the geodesic equation subjected to the boundary conditions of CTCs?
where the limits of $g(x)$ and $h(x)$ are easy to compute, thus allowing the original limit to be determined by taking the square root of this limit?
It is annoying when I have questions that is literally asking
"What is the question that will satisfy the given conditions"
because I have no idea how to quantify the plausibility of solving a question given a specific method
For example, if I have a question that asks "what is the function $f(x)$ such that its integral is most easily solved by squaring the integral?", we all knew $f(x)=e^{-x^2}$ (for starters)
but limits are not like integrals. Even the rules on solving and approaching limits are more art than solving integrals
(arghhh, you fill in the blanks for me for what my incoherent mind is trying to say, since you guys are somehow better and grasping what my mind is thinking)
...for the integral example, more like, we most commonly solve it by squaring it
Joke: Basically, I am tempted to prove that the solution to "what is the question that has the working solution as follows <workings>" is the same as "How to reach the tree of knowledge"
I have no idea why most of my questions are equivalent to this question
people like this guy should get a temp ban honestly. Just aggressively arguing with people giving constructive feedback. Unscientific and poor humility when asking physicists questions.
Denition 2.10. A spacetime (M; g) is called globally hyperbolic, if it satises
the strong causality condition and for all x; y 2 M the set J M
+ (x)\J Mô€€€ (y) is compact.
Can anyone tell me how does second condition implies, that no gravitational singularities without
an event horizon exist (als...
My understanding of some physical quantity being infinitesimal is some change that is so small that it causes no real change. It only gives a finite value on integration.
user116211
$\bf r$ is the position coordinate and about that the infinitesimal rotations took place; it's explicitly clear from the pic.
if a vector is infinitesimal, it means these vectors are small enough that we can regard them as basically lying on a plane at that point. Therefore the usual parallelgram rule applies and you can add them as if they live in flat space
I have studied about tensors up to order 3 but I have no intuition for axial vectors and I ignored these minuscule points in rotational kinematics in my class.
@YashasSamaga So that diagram of Frobenius is basically saying at every point along those circles, parallelgram rule is obeyed thus the tangential vectors dr1 and dr2 are indeed vectors
@Obliv I am guessing the OP is asking whether the big bang is caused by basically a universe filled with dark matter collided with a universe filled with dark energy
> 1+(f/,..II) will be denoted by 1+(f/), and is an open set, since ifp E..II can be reached by a future-directed timelike curve from f/ then there is a small neigh- bourhood ofp which can be so reached.
@YashasSamaga The disks are rotated by only a bit, e.g. maybe like 0.001 degrees each all at the same time, thus the sector traced out by that rotation on each disks is roughly a triangle. The diagram exaggerate this in order to make them visible
@Obliv Well, it is not entirely outlandish if you knew a bit about brane cosmology (which john or slereah can probably tell you more about it). In brane cosmology, one possible origin of the big bang is when two branes collided
I personally have drew many of those back in those time when I work with that model. Drawing cones help visualise the causal structure of the spacetime since you can easily see where they twist and do interesting things
@Slereah Oh lord Sachs-Wu has "show that the vacuum Einstein equations arise as the Euler-Lagrange equations of the functional $\int S\Omega$" as an exercise
how is a cartesian product of a group and a set written? $G \times A$ where $G$ is a group and $A$ is a set. Are elements $(g,a)$ with $g \in G$ and $a \in A$?
@Obliv A group is a special case of a set, I would expect $G \times A$ be a set in general as closure may be violated in general for the elements in the cartesian product
A group action of a group $G$ acting on a set $A$ is a map from $G \times A \to A$ (written as $ga$, for all $g \in G$ and $a \in A$) satisfying: $g_1 (g_2 a) = (g_1 g_2 )a$ ...
can someone explain what associativity is being satisfied here? Is this associativity between $G \times A$ and $G$? (I say that because $ga$ looks to be an element of $G \times A$)
All vectors, except $\:\mathbf{r}\:$, are infinitesimals.
I wonder if the author (Irodov) makes use of this result anywhere in his textbook.
