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

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SUMMARY

The discussion centers on the concept of linearity in the context of the subspace of T_p as described by Hawking and Ellis. Specifically, the subspace defined by the equation \langle \omega,x \rangle=(constant) is identified as linear, despite the constant being non-zero, which raises questions about the inclusion of the zero vector in T_p. The analogy drawn with Euclidean space illustrates that while the points r · a = d form a plane, they do not constitute a true subspace due to the shift introduced by the constant. This highlights the nuanced understanding required to grasp the linearity conditions in this mathematical framework.

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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 T_p defined by \langle \omega,x \rangle=(constant) for a given one-form \omega, is linear."

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

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

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

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