Lecture 9: Newtonian spacetime is curved! (International Winter School on Gravity and Light 2015)

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Strong rec Ben H · 2025-04 · Physics Stack Exchange · on “What does Newton's laws say about the geometry of spacetime?”

Always ask: WWFSS [LINK] (What would Frederic Schuller Say)? "Newtonian Spacetime is Curved!" [this video] (Lecture 9). This is a side lecture to his mathematical development of general relativity, which he does as a fun example just after introducing all of the main kinematic tools of differential geometry (manifolds, vector fields, covariant derivative / connection, parallel transport (geodesics) and curvature). I'll try to summarize his arguments briefly: A curved space (like the standard GR analogy of orbits: a marble rolling around a bowling ball on a trampoline) is impossible. Newton's Second Law for an object in a gravitational potential: $$ \ddot{x}^{\alpha} = f^\alpha (x) \quad \text{w/}

Recommends Christoph · 2016-05 · Physics Stack Exchange · on “Mathematical Formulation of Classical Spacetime”

I am particularly interested in a claim Schuller made in one of his classes. He said that the idea of having a metric on Newtonian spacetimes is an artifact of inertial reference frames. That's why in the first definition there is no metric, but instead a much weaker object, a connection. So, I've now watched parts of the video [this video] you posted in the comments, including the part I'm assuming you're basing this statement on. It's not about the derivation of a metric from the connection (ie metrizability), but the existence of a spatial metric. Quoting the video: Having a metric of space is an inertial system fiction Emphasis mine. What Schuller wanted to point out is that in non-inertial s

Recommends Ivan Burbano · 2016-05 · Physics Stack Exchange · on “Mathematical Formulation of Classical Spacetime”

Moreover, the first approach has physical interest because using it and expressing Newton's firs law in a slightly different language one can (staying on the classical regime) express Gravity not as a force but as a phenomena which changes what a straight line looks in spacetime. That is, as a curvature in spacetime. This leads to a field equation equivalent to Newton's gravity which looks very similar to Einstein's equations! This is a result Lagrange tried to obtain but failed because it only works in spacetime and with the right mathematics! [this video]

Recommends Qmechanic · 2018-04 · Physics Stack Exchange · on “Why can't Newton's 1st law be expressed as an autoparallel transportation in space?”

If you know GR [LINK] this question may irks you, since a relativistic point particle in a gravitational field do in fact follow geodesics (which is a special type of autoparallels [LINK] ) in spacetime. But the devil is in the details: Prof. Schuller is talking about autoparallels in space, not in spacetime. And he makes the case that the gravitational acceleration $\vec{g}$ (which depends on position, not velocity) cannot be emulated by a term quadratic in velocity, as needed in the autoparallel equation, cf. OP's eq. (1). Later at 36:06 [this video] in the same lecture 9 Prof. Schuller considers the same question in spacetime (as opposed to space ), and shows that a point particle in a gravita

Recommends Kajelad · 2020-10 · Mathematics Stack Exchange · on “Equivalence of curved space and Hamiltonian with potential”

Forces cannot, generally speaking, be encoded in a curvature of *space*, but they can (in some cases) be encoded in a curvature of space*time*. This lecture [this video] gives a pretty good overview of the correspondence.

Recommends G.Lang · 2021-02 · Physics Stack Exchange · on “Why does a stationary object start moving if there is no force acting on it in general relativity?”

Newton's first law states that an isolated object (on which no forces act) moves at constant velocity, which more specifically means along a straight line at constant speed. If we are not considering gravity as a force, but rather as a geometric constraint, then it turns out that this law can be applied to situations where particles move freely (with no other interactions) in a gravitational field (which is most relevant for your example), even in Newtonian mechanics, e.g. without relativity! As spacetime curves, the meaning of the terms "constant velocity" and "straight line" change, to reflect this curving. It turns out that the straight line is now the trajectory followed by your particle

Recommends amilton moreira · 2021-04 · Physics Stack Exchange · on “What is meaning of the phrase "The laws of physics are the same in all inertial frames of reference."?”

@ J. Murray That is because you are using a wrong formulation of Newtonian physics. Take a look at this lecture [this video]

Recommends asmaier · 2023-02 · Physics Stack Exchange · on “Newtonian gravity as curvature of space”

See this Lecture 9: Newtonian spacetime is curved! (International Winter School on Gravity and Light 2015): [this video]

Recommends S.G · 2023-10 · Physics Stack Exchange · on “What kind of coordinate change is needed to make gravity disappear?”

K.Pull asked: But then, acceleration has only three parameters, so how are they enough to make all the Christoffel symbols go to zero? One needs to keep in mind that the Christophel symbols [LINK] are defined in spacetime. As such, one needs to consider the acceleration also as something that has four components. The following might appear a lengthy response but it might be worth a read. For 3D spatial motion, we have the trajectory as (where $X$ is a chart-map a.k.a. coordinate system) $$X: \mathbb{R} \rightarrow \mathbb{R}^{3}$$ $$X: t \rightarrow \left(x^{1}(t), x^{2}(t), x^{3}(t)\right)$$ Now, for an accelerating particle, the equation of motion would be ( $\alpha$ runs from (1,3)) $$m \

Recommends Taylor Rendon · 2024-02 · Mathematics Stack Exchange · on “How is an autoparallel equation related to Newton's second axiom?”

I am currently working through The International Winter School on Gravity and Light [LINK] lecture series and have had some trouble understanding the following claim made in lecture $9$ at the 22:30 [this video] minute mark. Dr. Schuller says that asking " $(1)$ Can gravity be encoded in the curvature of space, such that its effects show if particles under the influence of (no other) force are postulated to move along straight lines in this curved space?" is the same as asking: " $(2)$ Is $$\ddot{x}^{\alpha}(t)-f^{\alpha}(x(t)) = 0$$ of the form of a autoparallel equation?" Note $\ddot{x}^{\alpha}$ is the ( $\alpha^{th}$ -component of the) acceleration of the test particle and $f^{\alpha}$ is the