Einstein's General Theory of Relativity | Lecture 12 | Summary and Q&A

Transcript

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Summary

In this video, the speaker discusses the concept of the accelerated reference frame in the context of black holes. They explain how observers in an accelerated reference frame move on hyperbolic trajectories, and how the acceleration increases as they move closer to the light cone. The video also explores the metric in polar coordinates, the analogy with ordinary polar coordinates, and the relationship between the accelerated frame and the Schwarzschild metric.

Questions & Answers

Q: Does special relativity not apply to accelerated frames?

Special relativity does not apply to accelerated frames in the same way as it does to inertial frames. In an accelerated reference frame, each observer is moving on a hyperbolic trajectory, and the acceleration increases as they move closer to the light cone. This is different from the situation in inertial frames where all observers experience the same acceleration.

Q: How does gravitational field vary in special relativity?

In special relativity, the closest thing to a uniformly accelerated reference frame already has the property that the gravitational strength or acceleration felt by an observer varies with distance from a particular point. This is similar to the variation of acceleration experienced by an observer moving closer to the center of the Earth in the Earth's gravitational field.

Q: What coordinates characterize different observers in the accelerated reference frame?

Observers in the accelerated reference frame can be characterized by their distance from the origin of coordinates (denoted as "r") and the hyperbolic angle ("omega"). In terms of the original coordinates on the blackboard ("x" and "t"), the relationship is expressed as "x = r * cosh(omega)" and "t = r * sinh(omega)". These coordinates resemble polar coordinates, but with the use of hyperbolic functions instead of trigonometric functions.

Q: What is the metric in polar coordinates in special relativity?

In polar coordinates in special relativity, the metric becomes "ds^2 = dr^2 - r^2 dtheta^2", where "ds" is the proper distance, "dr" is the change in radial coordinate, and "dtheta" is the change in angular coordinate. This metric captures the essence of the geometry in a two-dimensional space.

Q: Is there a connection between the accelerated reference frame and the Schwarzschild metric?

Yes, there is a connection between the accelerated reference frame and the Schwarzschild metric. By analyzing the Schwarzschild metric, one can gain insights into the behavior of black holes and their horizons. The Schwarzschild metric describes the geometry in the vicinity of a spherically symmetric mass, and is a general solution of Einstein's equations for the gravitational field outside the mass. The connection between the two allows for the exploration and understanding of various phenomena associated with black holes.

Q: How can we approximate the Schwarzschild metric near the black hole horizon?

Near the black hole horizon, we can approximate the Schwarzschild metric by considering a small neighborhood of the horizon and making certain assumptions. By setting "r" to be close to 2 times the mass of the black hole (denoted as mg), we can simplify the metric by replacing the terms "d_rho^2" and "d_omega^2" with "dy^2 + dz^2". This approximation is valid in the immediate vicinity of the horizon and allows us to better understand the geometry and properties of the black hole.

Q: Why is it possible to replace the metric terms with a tangent plane approximation near the horizon?

Near the horizon, the geometry of the black hole can be approximated by a tangent plane. This is because the immediate neighborhood of a point on a sphere, such as the black hole horizon, looks like a flat plane. By considering an infinitesimally small region around a point on the horizon, we can simplify the metric terms to be like those of a flat plane, allowing for a better intuitive understanding of the black hole's behavior.

Q: How can we further simplify the metric near the black hole horizon?

To further simplify the metric near the black hole horizon, we can introduce a new variable called "rho" which represents the proper distance from the horizon to an arbitrary point. By equating "rho" to the square root of "r - 2mg", we can rewrite the metric in terms of "rho" and obtain a metric that closely resembles the tangent plane approximation. This change of variables helps to clarify the geometric properties of the black hole near its horizon.

Q: What is the proper distance between the black hole horizon and an arbitrary point?

The proper distance between the black hole horizon and an arbitrary point "r" is equal to twice the square root of "r - 2mg" divided by the square root of 8mg. This proper distance represents the spatial distance from the horizon to the point of interest and can be used as a new variable to simplify the metric and analyze the geometry near the horizon.

Q: How does the metric change when expressed in terms of the proper distance "rho"?

When expressed in terms of the proper distance "rho," the metric near the black hole horizon becomes simpler and closely resembles a flat space metric. The numerator term of the metric becomes "rho^2 / 8mg," while the denominator term remains the same. This change of variables helps to highlight the geometry and behavior of the black hole in the vicinity of its horizon.

Q: What have we learned about the black hole horizon from the analysis so far?

Through the analysis of the metric near the black hole horizon, we have gained insights into the behavior and geometry of the black hole in its immediate vicinity. By using suitable coordinate transformations, we can approximate the metric as a flat space metric and study the properties of the black hole near its horizon. This analysis allows us to better understand the effects of gravity and the curved space-time around black holes.