By Eduard Alexis Larrañaga R.

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1. HIPERSUPERFICIES NULAS 45 n = ev/4M ∂v . 42) Ejemplo. 44) gϕϕ = r sin θ. 45) Consideraremos las hipersuperficies definidas por la función Φ = U . La hipersuperficie nula r = 2M corresponde a Φ = U = 0. El vector normal a estas hipersuperficies esta definido por n = N g U σ ∂U Φ∂σ = N g U V ∂V r = −N er/2M ∂V . 16M 3 En este caso, al evaluar el vector normal en la hipersuperficie nula r = 2M se tiene n = −N e ∂V . 50) y ya que es idénticamente igual a cero, entonces ∂ µ n2 = 0. 51) nσ ∇σ nµ |N = 0.

78) cuando xi −→ ∞ (infinito espacial) o cuando x0 = t −→ ∞ (infinito temporal). Para comprender como funciona esta técnica tomaremos como primer ejemplo la variedad de Minkowski y la de Rindler, para luego realizar la compactificación de Kruskal. 1. Espacio-Tiempo de Minkowski El espacio de Minkowski en coordenadas polares se puede escribir como 32 CAPÍTULO 2. 79) donde las coordenadas t y r tienen los rangos −∞ < t < ∞ 0 ≤ r < ∞. 81) Es importante notar que la singularidad en r = 0 es una singualridad coordenada como puede comprobarse al pasar a otro sistema (por ejemplo a coordenadas cartesianas).

45) Consideraremos las hipersuperficies definidas por la función Φ = U . La hipersuperficie nula r = 2M corresponde a Φ = U = 0. El vector normal a estas hipersuperficies esta definido por n = N g U σ ∂U Φ∂σ = N g U V ∂V r = −N er/2M ∂V . 16M 3 En este caso, al evaluar el vector normal en la hipersuperficie nula r = 2M se tiene n = −N e ∂V . 50) y ya que es idénticamente igual a cero, entonces ∂ µ n2 = 0. 51) nσ ∇σ nµ |N = 0. 53) ∂ . 54) N =− y con ello el vector normal será simplemente n= 46 CAPÍTULO 3.

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