Recent content by VladZH

So basically if solution did exist that would mean the metric is flat?

Thanks for reply I understand the idea that connection can be zero at one point but not in others. But how I can derive it from the equation I wrote? Since that transformation will turn metric components to diagonal in any point. I just want to find an error in my steps

That is what I meant. Given coordinates xi and metric components gij I find such cordinates x'i=f(x1,...,xn) so that having Jacobian J_j^i=\frac{\partial x_i}{\partial x'_j} I am getting new metric components J^T*g*J=g'= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} Than I solve geodesic...

So how coordinate transforamtion is different from the example in my first post? For coordinate transform of metric tensor we have J^T*g'*J=g where J is Jacobian matrix.

What is the difference? What are vector components then?

I cannot fully understand what you mean. If we want to know vector components x^i in another coordinate system we can use equation x'^j=B^j_i x^i And for components of metric tensor respectively g'_{ij}=A^l_iA^k_j g_{kl} where A = J and B = J^{-1}. Why it is not coordinate transformation if we...

So in this specific example can I say that g' is equivalent to euclidean metric? For coordinate transformation I can use A-1, right?

Thanks for reply! Yes, I mean "equation of the form y = ax + b" . The solution of geodesic equation in coordinate system with component of metric as identity matrix gives y = ax + b. So I can take any metric in the coordinate system where its components are identity matrix and solving...

I will start with an example. Consider components of metric tensor g' in a coordinate system $$ g'= \begin{pmatrix} xy & 1 \\ 1 & xy \\ \end{pmatrix} $$ We can find a transformation rule which brings g' to euclidean metric g=\begin{pmatrix} 1 & 0 \\ 0 & 1\\ \end{pmatrix}, namely...

This is problem for my video game I tried to solve a simpler problem when we don't have the body C. Let P(r, φ) is a point on the circle. Let s between A and P. Hence, the time for spacecraft from A to B equals Δt=s/v The time for body B to get P is Δt=Δφ/ω. We get d/v=Δφ/ω where φ=sω/v Now...

Hello Given: Point A Body B with angular velocity ω C body with radius r Spacecraft with constant velocity v. We neglect the gravity of the bodies B, C The problem: Find the shortest trajectory for spacecraft from A to B What approaches might be here? How might the solution be changed if...

Thank you, guys. Seems like I confused with the formultaion

I'm trying to proof an identity from Munkres' Topology A \ ( A \ B ) = B By definition A \ B = {x : x in A and x not in B} A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B What did I miss?

Ok, I see my approach is wrong. What are the approches to show that change of mass affects the curvature and change of velocity does not? How can we use Einstein field equation here?

Sorry. I'm not talking about covariance