Conversation started Jan 22, 2021 at 22:53.
Charlie
Charlie
Jan 22, 2021 22:53
regarding the induced metric on the worldsheet in string theory, along which map do we pull-back the background spacetime metric?
ACuriousMind
ACuriousMind
@Charlie along the coordinates of the string, usually denoted $X$
Charlie
Charlie
hmm ok
ACuriousMind
ACuriousMind
you don't sound convinced :P
Charlie
Charlie
Jan 22, 2021 23:10
I guess it's odd to me that we pull back a flat metric onto flat spacetime
ACuriousMind
ACuriousMind
I'm not sure what that means - you're pulling back a flat spacetime metric onto a worldsheet that a priori has no metric
Charlie
Charlie
but is the worldsheet not just like, a submanifold of spacetime?
ACuriousMind
ACuriousMind
The $X$ makes it one
Charlie
Charlie
If we want to measure distances/areas on the worldsheet, why not just do it in a chart? it seems like the string is being treated as something that isn't just a sheet sitting in spacetime
ACuriousMind
ACuriousMind
The worldsheet is a 2d manifold $\Sigma$ and the coordinates $X : \Sigma \to\mathbb{R}^{26}$ are an embedding that identifies it as a submanifold
@Charlie I think you're trying to imbue the mathematics with a bit too much ontology here
Maybe an analogy helps: When we consider a curve in a manifold, the curve is not just "a submanifold of dimension 1", it is an embedding of the abstract manifold "the unit interval" into the manifold, i.e. an embedding $\gamma : I \to M$. Likewise, a worldsheet is an embedding of a 2d manifold into another manifold
you wouldn't ask "why are we treating the curve as an interval that's not sitting in spacetime", would you?
Charlie
Charlie
Jan 22, 2021 23:20
oh, is the metric begin pulled all the way back to the 2d manifold itself?
that actually would make more sense
my problem was that it seemed trivial to give an submanifold a metric obtained by pulling back the metric of the surrounding space to it
ACuriousMind
ACuriousMind
since the 2d manifold is isomorphic to its image under the embedding, I'm not sure that's a meaningful question :P
The reason why "pulling back" a metric is the natural thing to do is precisely because the pulled back metric gives the same results as "restricting" the metric to the submanifold would
Charlie
Charlie
ohh
If it's the same as just restricting the metric that makes sense, both Tong's notes and my lecturer made a point of saying it's the pullback of the metric, which made me think there was something special about it
a bit like taking a circle, then considering a small part of the circle and measuring distances on it by just using the metric on the surrounding circle :p
that seemed like a strange point to labour
ACuriousMind
ACuriousMind
it's just standard differential geometry, but for some reason some physicists feel they suddenly need to pretend to do actual math when doing string theory (and then still not do it :P)
Charlie
Charlie
oh god wait the "coordinates" $X$ we're talking about here go from the 2d manifold into spacetime, right? We're not talking about manifold coordinates in the sense of charts
or hm
ACuriousMind
ACuriousMind
Jan 22, 2021 23:36
@Charlie yes, they're maps $X: \Sigma \to M$ where $M$ is the spacetime.
Charlie
Charlie
you've written the coordinates as $X:\Sigma\rightarrow \Bbb R^{26}$, is $\Bbb R^{26}$ the Euclidean space that is the target space of charts on spacetime or is it spacetime itself?
ah
now it seems like a slightly less trivial thing to do, because $\Sigma$ is now a separate manifold
ACuriousMind
ACuriousMind
but when the physicists write $X^\mu(\sigma, \tau)$, that of course is $X$ "expressed in charts" - the $(\sigma,\tau)$ live in some $\mathbb{R}^2$ chart of $\Sigma$ and the $X^\mu$ values live in some $\mathbb{R}^{26}$ chart of $M$
it's common not just in physics to not care much about the distinction between expressions "in local coordinates" and the abstract map, don't worry about that too much
Charlie
Charlie
I seee
ok ty, that is clearer
 
Conversation ended Jan 22, 2021 at 23:42.