Understanding "Linearity" in Hawking & Ellis' Subspace of T_p

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In summary, Hawking and Ellis discuss a subspace of T_p defined by \langle \omega,x \rangle=(constant) for a given one-form \omega, which is linear. However, there is confusion about the linearity of this subspace when the constant is non-zero. The analogy to euclidean space is mentioned, but it is noted that this subspace is not technically a subspace of the original space. The speaker questions if they are missing something, but it is confirmed that they are not. The following sentence emphasizes this. This concept is just one of the difficulties faced while reading this book.
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Manicwhale
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I'm reading Hawking and Ellis, and on p.16 they say that

"The subspace of [tex]T_p[/tex] defined by [tex]\langle \omega,x \rangle[/tex]=(constant) for a given one-form [tex]\omega[/tex], is linear."

But in what sense is this true? For if the constant is non-zero, the 0 of [tex]T_p[/tex] is not in the subspace, nor does it satisfy the usual linearity condition.

By analogy with euclidean space, the points [tex]r \cdot a = d[/tex] form a plane, but not technically a subspace of the original space (since it is "shifted"). Am I missing something?
 
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  • #2
No, I don't think that you are missing anything, especially considering the sentence that follows the sentence which you quoted.
 
  • #3
Thanks!

Turns out that's the least of my difficulties in this book, but it's quite interesting.
 

What is linearity in Hawking & Ellis' Subspace of T_p?

Linearity in Hawking & Ellis' Subspace of T_p refers to the property of a mathematical function or transformation that follows the principles of linear algebra, where the output is directly proportional to the input. In this context, it refers to the behavior of spacetime in the vicinity of a point p, where the curvature of spacetime can be described by linear transformations.

How does linearity affect the understanding of spacetime in Hawking & Ellis' Subspace of T_p?

Linearity plays a crucial role in understanding the behavior of spacetime in Hawking & Ellis' Subspace of T_p. It allows for the calculation of the curvature of spacetime in the vicinity of a point p, which is essential in understanding the dynamics of gravity and the behavior of matter in that region.

What are the implications of non-linearity in Hawking & Ellis' Subspace of T_p?

The presence of non-linearity in Hawking & Ellis' Subspace of T_p can have significant implications for our understanding of gravity and the behavior of matter in that region. It can lead to the formation of singularities, which are points where the curvature of spacetime becomes infinite and our current laws of physics break down.

How is the concept of linearity related to Einstein's General Theory of Relativity?

Einstein's General Theory of Relativity is based on the principle of general covariance, which states that the laws of physics should be the same for all observers, regardless of their reference frame. The concept of linearity is essential in this theory as it allows for the description of the curvature of spacetime, which is the foundation of gravity according to this theory.

Can linearity be applied to other areas of physics besides spacetime in Hawking & Ellis' Subspace of T_p?

Yes, linearity is a fundamental concept in many areas of physics, including classical mechanics, electromagnetism, and quantum mechanics. It allows for the description of the behavior of physical systems and is a crucial tool in solving complex problems and making predictions about the behavior of matter and energy in the universe.

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