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I think in your question, as currently formulated, the whole rotating cosmic string is a red herring. If I interpret your notation correctly, $a$ and $\kappa$ are constants. And hence locally you can …

Concerning the Lebesgue covering dimension, absolutely nothing can be said, if you work with manifolds of total space-time dimension 3 or higher. Preamble We will consider spacetimes with closed timel …

I would like to argue that the situation considered in the comments is "close to generic". Let $(M,g)$ be a Lorentzian manifold that is not strongly causal; this implies that $(M,g)$ is also not stabl …

If Lorentz invariance is not required: Let $\phi$ be any smooth bump function $\phi:\mathbb{R}\to\mathbb{R}$ that is non-zero precisely on $(0,4m^2)$ (including the one you used in the question statem …

In 2 dimensions the question posted in this comment has a negative answer. Consider the standard Minkowski space with double null coordinates $(u,v)$ in which the metric is $ds^2 = - du~dv$. Any stric …

Klein-Gordon equation has finite speed of propagation, which implies that if two solutions have initial data agreeing on the set $\{|\vec{x}| < t_2\}$, then the two solutions agree on the set $\{|x^0| …

Q1 The topology on $\mathbb{R}^4$ is the usual one. This is the general case for Lorentzian geometry: the topology is the one defined by the charts in your atlas. Q2 Given a fixed Lorentz transformati …

Yes. (Body of answer must be 30 characters, and I only entered four. So here are a bit more.) The Computation: On $\mathbb{R}^3$, set $y = \lambda x$ and so $dx = \lambda^{-3} dy$. Set $s = \lambda^ …

Write $z = e^{i2\theta}$ where $\theta$ is as in your second formulation, you have that the equation is equivalent to $$ -2i \partial^2_{xy} (z - \bar{z})+ (\partial^2_{xx} - \partial^2_{yy})(z + \bar …

I answered about the incompleteness theorem in the other thread. Let's talk about some of his other contributions here. (This list is definitely incomplete*, but just some stuff off the top of my head …

For the free evolution specifically, in relation to Mateusz's comment: on the Fourier side you can write the solution as (for $m = -1/2$; you can rescale/invert time to get other scalings) $$ \phi(t …

The short answer is "what physicists mean by 'warped' and 'twisted' geometry" is not the same as "what differential geometers mean by 'warped' and 'twisted' geometry". The use is a lot more qualitativ …

I think this just follows from linearity. The condition that $u(\sigma, q) = u(q,\sigma)$ implies that $P u(\sigma,\cdot) = 0$. In fact, you have that $q\mapsto u(\sigma,q)$ is the unique solution t …

Your question can be equivalently phrased as: Given a closed two-form $F$ on $\mathbb{R}^4$, is it always possible to find a Lorentzian metric on $\mathbb{R}^4$ such that $\delta F = 0$. Then t …

To make my comment more concrete: if you define $$\mathcal{G}_0 = \phi $$ and $$ \mathcal{G}_{n+1} = \Box \mathcal{G}_n - \frac{1}{n+1} \partial (\partial\cdot \mathcal{G}_n) + \frac{1}{(n+1)(2n+1)} …