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Hey guys. I'm having doubts about something in a GR text I read (Gravitation - T.Padmanabhan). In general, for Lorentzian 4-manifolds, we call a basis of smooth vector fields ##(e_{\mu})^{a}## (that is, a set of smooth vector fields that define a basis for the tangent space at each event of space-time) a tetrad if ##(e_{\mu})^{a}(e_{\nu})_{a} = \eta_{\mu\nu}## i.e. ##(e_{0})^{a}(e_{i})_{a} = 0,(e_{0})^{a}(e_0)_{a} = -1, (e_{i})^{a}(e_{j})_{a} = \delta_{ij}## so the basis of smooth vector fields are also orthonormal. In Padmanabhan's text, when defining Fermi-Walker transport, he says (I tweaked the notation a bit) "Consider an observer who is moving along some worldline (which is not necessarily a geodesic) with the 4-velocity ##u^{a}##. Let ##(e_{\mu})^{a}## be a tetrad of basis vectors transported along the trajectory of this observer...The first condition we will impose is that the basis should satisfy the standard relations, ##(e_{\mu})^{a}(e_{\nu})_{a} = g_{\mu\nu}##."
This seems to go against the very definition of a tetrad (as an orthonormal basis for the tangent space at each point on the observer's worldline) because the way it is defined in this text, the basis vectors need not be orthonormal. What gives? He later says that Fermi-Walker transport ensures that the tetrad remains orthonormal from the initial erected tetrad (which is of course true in a general context) but he doesn't even define it to be orthonormal to start with so what's up with that?!
My next question involves the use of the term "local Lorentz frame / local inertial frame" in GR. Here is the issue: consider an observer moving along an arbitrary worldline and let ##p## be an event on the observer's worldline. Now by a procedure much like the Gram-Schmidt scheme, we can always find a basis ##(e_{\mu})^{a}|_p## for ##T_p M## such that ##g_{\mu\nu}|_p = \eta_{\mu\nu}## with respect to this basis i.e. we can always find a point-wise tetrad ##(e_{\mu})^{a}|_p## for ##T_p M##. Now I've seen some sources (including Padmanabhan) call this point-wise tetrad a local Lorentz frame / locally inertial reference frame for the observer at ##p## which I find incorrect because from what I've learned, the concept of a locally inertial reference frame at ##p## requires that the frame be derived from a coordinate system ##\{x^{\mu}\}##, which we call locally inertial coordinates, that the observer must setup in a neighborhood ##U## of ##p## (and when I say the frame is derived from ##\{x^{\mu}\}## I mean ##(e_{\mu})^{a}|_p = (\partial_{\mu})^{a}|_p##) such that ##\partial_{\alpha}g_{\mu\nu}|_p = 0## (this further ensures that the metric tensor is approximately the Minkowski metric for any ##q\in U## to first order).
This physically ensures that the locally inertial reference frame at ##p## is that of a freely falling observer (because it is precisely the freely falling observers who are locally inertial) i.e. ##u^{a}\nabla_{a}u^{b}|_p = (e_{0})^{a}\nabla_{a}(e_{0})^{b}|_p = (\partial_{0})^{a}\nabla_{a}(\partial_{0})^{b}|_p = \Gamma^{b}_{ac}|_p (\partial_{0})^{c}(\partial_{0})^{a}|_p = 0##
where I have used the fact that in any reference frame of an observer at an event ##p## on his/her worldline, the 4-velocity ##u^{a}|_p = (e_0)^a|_p##.
So you can see why I am confused when Padmanabhan, and others, call the tetrad ##(e_{\mu})^{a}|_p## at an event ##p## on an arbitrary observer's worldline (i.e. a point-wise tetrad which is not necessarily derived from a locally inertial coordinate system about ##p##) a local Lorentz frame / locally inertial frame since no restriction is placed on the observer being in free fall. What am I missing here? Thanks in advance for any help.
This seems to go against the very definition of a tetrad (as an orthonormal basis for the tangent space at each point on the observer's worldline) because the way it is defined in this text, the basis vectors need not be orthonormal. What gives? He later says that Fermi-Walker transport ensures that the tetrad remains orthonormal from the initial erected tetrad (which is of course true in a general context) but he doesn't even define it to be orthonormal to start with so what's up with that?!
My next question involves the use of the term "local Lorentz frame / local inertial frame" in GR. Here is the issue: consider an observer moving along an arbitrary worldline and let ##p## be an event on the observer's worldline. Now by a procedure much like the Gram-Schmidt scheme, we can always find a basis ##(e_{\mu})^{a}|_p## for ##T_p M## such that ##g_{\mu\nu}|_p = \eta_{\mu\nu}## with respect to this basis i.e. we can always find a point-wise tetrad ##(e_{\mu})^{a}|_p## for ##T_p M##. Now I've seen some sources (including Padmanabhan) call this point-wise tetrad a local Lorentz frame / locally inertial reference frame for the observer at ##p## which I find incorrect because from what I've learned, the concept of a locally inertial reference frame at ##p## requires that the frame be derived from a coordinate system ##\{x^{\mu}\}##, which we call locally inertial coordinates, that the observer must setup in a neighborhood ##U## of ##p## (and when I say the frame is derived from ##\{x^{\mu}\}## I mean ##(e_{\mu})^{a}|_p = (\partial_{\mu})^{a}|_p##) such that ##\partial_{\alpha}g_{\mu\nu}|_p = 0## (this further ensures that the metric tensor is approximately the Minkowski metric for any ##q\in U## to first order).
This physically ensures that the locally inertial reference frame at ##p## is that of a freely falling observer (because it is precisely the freely falling observers who are locally inertial) i.e. ##u^{a}\nabla_{a}u^{b}|_p = (e_{0})^{a}\nabla_{a}(e_{0})^{b}|_p = (\partial_{0})^{a}\nabla_{a}(\partial_{0})^{b}|_p = \Gamma^{b}_{ac}|_p (\partial_{0})^{c}(\partial_{0})^{a}|_p = 0##
where I have used the fact that in any reference frame of an observer at an event ##p## on his/her worldline, the 4-velocity ##u^{a}|_p = (e_0)^a|_p##.
So you can see why I am confused when Padmanabhan, and others, call the tetrad ##(e_{\mu})^{a}|_p## at an event ##p## on an arbitrary observer's worldline (i.e. a point-wise tetrad which is not necessarily derived from a locally inertial coordinate system about ##p##) a local Lorentz frame / locally inertial frame since no restriction is placed on the observer being in free fall. What am I missing here? Thanks in advance for any help.
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