Smooth deformation of a Lorentzian manifold and singularities

• MeJennifer
In summary, Schoen and Yau addressed the issue of how a smooth deformation of a Lorentzian manifold can lead to one or more singularities in their paper on the positivity of mass in general relativity. They used asymptotically flat initial data and solved the problem, which was published over two decades ago. The paper is highly technical and difficult to understand, but the authors provide an outline of their thinking.
MeJennifer
How can a smooth deformation of a Lorentzian manifold possibly create one or more singularities?

MeJennifer said:
How can a smooth deformation of a Lorentzian manifold possibly create one or more singularities?

Schoen and Yau solved this problem over two decades ago in the context of asymptotically flat initial data in the last of their three famous papers on the positivity of mass in general relativity. The reference is

@ARTICLE{1983CMaPh..90..575S,
author = {{Schoen}, R. and {Yau}, S.-T.},
title = "{The existence of a black hole due to condensation of matter}",
journal = {Communications in Mathematical Physics},
year = 1983,
month = dec,
volume = 90,
pages = {575-579},
adsnote = {Provided by the Smithsonian/NASA Astrophysics Data System}
}

Beware, the paper is hugely technical. I can come up with a broad outline of their thinking if you want, but it's still going to be hard to understand.

Just pinch one point and lift it up to infinity.

1. What is a Lorentzian manifold?

A Lorentzian manifold is a type of mathematical space that is used to describe curved spacetime in the theory of general relativity. It is a four-dimensional space that includes both space and time dimensions, and it is characterized by a metric that follows the principles of special relativity.

2. What does it mean for a Lorentzian manifold to have smooth deformation?

Smooth deformation refers to a continuous change in the shape or structure of a Lorentzian manifold. This means that the manifold can be deformed in a way that preserves its smoothness, without any abrupt changes or discontinuities. This is an important concept in the study of curved spacetime and allows for a better understanding of the behavior of matter and energy in the universe.

3. What are singularities in the context of Lorentzian manifolds?

Singularities are points in a Lorentzian manifold where the curvature becomes infinite. These points represent the breakdown of the laws of physics and are often associated with extreme conditions, such as the center of a black hole. The study of singularities is important in understanding the behavior of the universe at its most extreme.

4. How does smooth deformation affect singularities in a Lorentzian manifold?

Smooth deformation can help to explain and potentially resolve singularities in a Lorentzian manifold. By studying how the manifold can be smoothly deformed, scientists can gain insight into the nature of singularities and potentially find ways to avoid them. It is also possible that smooth deformation could lead to new theories or models that can better describe the behavior of the universe at these extreme points.

5. What are some real-world applications of studying smooth deformation of Lorentzian manifolds and singularities?

The study of smooth deformation of Lorentzian manifolds and singularities has many practical applications. It can help in predicting and understanding the behavior of objects in extreme environments, such as black holes. It can also aid in the development of new theories and models for understanding the universe. Additionally, the study of smooth deformation can have implications for engineering and design, as it can provide insights into the behavior of materials under extreme conditions.

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