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Could someone please tell me what is a quasi-local charge? For instance, why are Brown-York charges called quasi-local?

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The value of a local quantity at a particular point in spacetime can only depend on the values of other quantities at that same point in spacetime. It cannot depend on the values of those quantities at other points in spacetime.

In contrast, the value of a nonlocal quantity at a particular point in spacetime can depend on the values of other quantities at other points in spacetime. There is nothing in principle preventing a nonlocal quantity from depending on the value of other quantities at every point in spacetime, out to infinity.

Quasi-local quantities forbid some types of nonlocal behavior. In particular, a quasi-local quantity is only allowed to depend on the values of other quantities in a finite region of spacetime around that point.

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  • $\begingroup$ Thx for your answer. So for instance in the case of the BY quasilocal charge given by $$\int_\Sigma \mathrm{d}^3x \sqrt{\sigma} u_a T^{ab} \xi_b$$, what are the local and non-local quantities that it depends on? $\endgroup$
    – Y2H
    May 11, 2020 at 15:20
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    $\begingroup$ @Y2H You can see all of the quantities it depends on in the integrand. You integrate these quantities over some finite region $\Sigma$, which makes the charge quasilocal, since the charge depends on the values of those quantities in a finite spacetime region. $\endgroup$
    – probably_someone
    May 11, 2020 at 16:48

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