2025년 7월 4일 금요일 아카이브

• Light Cones: How Spacetime Organizes the Causality of the Universe

  작성자: roberto.c.alfredo 방: physics 2025. 7. 4. 오후 6:42에

  1. What Exactly Are Light Cones?

  When you turn on a flashlight in a dark room, the light spreads out in a cone of illumination. Now imagine that effect on a cosmic scale with the universal speed limit: light itself. In relativity, these cones mark the boundaries of the causal past and causal future of every event.

  Formally, an event is just a point in spacetime with coordinates \((t, x, y, z)\). The light emitted from that point travels along surfaces that form cones, clearly separating which events can influence it or be influenced by it.

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  2. Simple Mathematical Structure of the Light Cone

  Recall our spacetime interval:   $$ s^2 = c^2 t^2 - x^2 - y^2 - z^2 $$   The light cones are the regions satisfying:

  • \(s^2 = 0\): lightlike trajectories.
  • \(s^2 > 0\): inside the cone, timelike trajectories.
  • \(s^2 < 0\): outside the cone, spacelike trajectories.

  Visually, these cones are the surfaces traced out by light emitted at the central event, neatly partitioning spacetime into causal regions.

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  3. Past, Future, and Causality: What’s Causally Possible?

  • Causal past (lower cone): all events that could have influenced the present.
  • Causal future (upper cone): all events that can be influenced by the present.
  • Non-causal region (outside the cone): events that can neither influence nor be influenced, since doing so would require exceeding \(c\).

  This is the physical guarantee of cause-and-effect order in the universe. No observer will ever see events flipped in time, avoiding classic time-travel paradoxes.

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• Effet Doppler relativiste et aberration de la lumière

  작성자: roberto.c.alfredo 방: physics 2025. 7. 4. 오후 7:16에

  1. Du sifflet d’ambulance au cosmos : qu’est-ce que mesure l’effet Doppler ?

  En acoustique, on entend la montée de la fréquence quand une ambulance arrive et sa descente quand elle s’éloigne.   En relativité, l’effet Doppler s’applique à la lumière, pas au son, et la formule classique ne suffit plus.

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  2. Doppler longitudinal : fréquence et longueur d’onde

  Pour un émetteur et un récepteur s’éloignant ou se rapprochant selon la ligne de visée :   $$ \frac{\lambda_{\text{obs}}}{\lambda_{\text{em}}} = \sqrt{\frac{1 + \beta}{1 - \beta}}, \quad \beta = \frac{v}{c}. $$   En termes de fréquences :   $$ \nu_{\text{obs}} = \nu_{\text{em}}\, \sqrt{\frac{1 - \beta}{1 + \beta}}. $$

  • Si \(v>0\) (éloignement) : décalage vers le rouge (\(\lambda_{\text{obs}}>\lambda_{\text{em}}\)).
  • Si \(v<0\) (approche) : décalage vers le bleu (\(\lambda_{\text{obs}}<\lambda_{\text{em}}\)).

  Exemple astronomique :   Une planète en orbite à 30 km/s provoque un décalage d’environ 0,01 nm sur une raie de fer, suffisant pour sa détection.

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  3. Aberration de la lumière : changement d’angle apparent

  L’aberration modifie l’angle \(\theta\) entre un rayon lumineux et la direction du mouvement. Si \(\theta\) est mesuré par la source et \(\theta'\) par l’observateur :   $$ \cos\theta' = \frac{\cos\theta - \beta}       {1 - \beta\,\cos\theta}. $$

  • Pour \(\theta=90^\circ\) : les étoiles semblent “poussées” vers la direction du déplacement terrestre, jusqu’à un maximum d’environ \(20{,}5''\).
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