# Questions tagged [differential-geometry]

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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### Confusion on number of component of Cauchy stress tensor

The Cauchy stress tensor is often presented as a tensor having $(2,0)$ tensor having nine components in any given basis. However, I think it should actually be $6 \times 3 =18$ because a cube has six ...
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### Where did Henry Poincare state that we should not sum linear movements with curvilinear movements?

Years ago, I was arguing with a P.h.d. in Relativity, and he showed me a print screen that Henri Poincare stated that we should not sum curvilinear movements with linear ones. But, I lost that print ...
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### Physical significance of metric compatibility

When we try to construct a covariant derivative, we impose several conditions on it so that the resulting derivative is unique. However, I can't make sense of the condition of metric compatibility. I ...
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### Geodesic deviation and Lie dragging

Suppose that $𝑥_\mu(\lambda,𝑠)$ represents a family of curves. Let $𝑣_𝜇$ represents the the tangent vector to a curve $𝑥_𝜇(\lambda,𝑠_0)$ with $𝑠_0$fixed that is $𝑣_𝜇=∂𝑥_𝜇/∂\lambda$ and ...
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### Kerr Solution metric ansatz for EFEs

The Schwarzschild metric ansatz is given by $$ds^2=-A(r)dt^2+B(r)dr^2+r^2d \Omega^2$$ where upon applying the Einstein Field Equations $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we obtain the normal ...
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### What is a signature of pp-wave metric?

pp-wave spacetime metric is defined in Brinkmann coordinates as $$ds^2 = H(u,x,y)du^2 + 2 du dv + dx^2 + dy^2.$$ Since it's lorentzian (https://en.wikipedia.org/wiki/Pp-wave_spacetime), I wonder what ...
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### How to find a complete understanding the 2nd law of Thermodynamics in terms of forms?

I have two straightforward question, and below I introduce more context to interpret them: What is, or is there, an order relation for forms that one can use to make sense of the 2nd law of ...
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### What is the idea behind 2-spinor calculus?

In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
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