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On Professor Frederic P. Schuller's Lecture about General relativity, where you can access it through this link: https://www.youtube.com/watch?v=IBlCu1zgD4Y, he clamed Newton's axioms can be converted into 3 mathematical axioms. However I don't understand how they are connected.

Definition: A Newtonian spacetime is a quintuple $(M,O,A,∇,t)$ where $(M,O,A,∇)$ is a differentiable manifold with a torsion free connection, and $t\in C^\infty$M is a smooth function such that $(dt)_p\neq 0$ and $\nabla(dt)=0$.

He claimed that "There is an absolute space" can be written as $(dt)_p\neq 0$. And "Absolute time flows uniformly" can be written as $\nabla(dt)=0$ everywhere. How should I understand these restriction?

user45664
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3 Answers3

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Did you find the answer to your question?

The second one $\nabla(dt)=0$ implies that the $\Gamma^0_{ab}=0$. Using Newton's second law $$\nabla_{v_x} v_x=\frac{F}{m}$$ and the fact that the force field $F$ is spatial i.e. $dt(F)=0$, leads to a linear equation in absolute time $t$ and the parameter $\lambda$ that parameterizes the worldline. I think this is what he meant by time flows uniformly.

The first one I'm not so sure, $(dt)_p= 0$ implies that the time component of every vector (here you may imagine a vector as velocity at point $p$ on the curve $\gamma(\lambda)$ at $\gamma(\lambda_0)=p$) is zero. "Intuitively" this means that there is a point $p\in M$ such that all the curves passing through $p$ have no velocity component in time direction implying there exists segments on the curve $\gamma$ where time is constant as a function of the parameter $\lambda$ contradicting our previous assumption. I don't know why he refers to this as "There is an absolute space".

Please let me know if you have a better explanation.

Razor
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I think I sort of understand the $\nabla (dt)$ condition.

Suppose we have a room and consider any point in the room, in the Newtonian case, it must be that if you kept a clock at each point, then , say if $x$ seconds passed on one clock, in all other clock that same time interval should pass.

I am not quite sure how one would interpret the equation when talking about how the time one form changes when we take covariant derivative in the time basis direction...

tryst with freedom
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$\nabla (dt) = 0$, means that given all clocks are synchronized, there's no time flow gradient going from point to point in system, or in other words : $$ \frac {\partial (\Delta t)}{\partial x} + \frac {\partial (\Delta t)}{\partial y} + \frac {\partial (\Delta t)}{\partial z} = 0 $$ This means that time flow is absolute in Newtonian axioms,- it does not depend on points relative speed difference (otherwise, special relativity), nor on gravitational field potential difference between the points being analyzed (otherwise, general relativity).

Agnius Vasiliauskas
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  • Nice, this is a mathematical abstraction of what I said. Still, I have a question here. The lecturer considers the space we are working in as $(t,x,y,z)$ so the gradient also considers a time derivative part. I can't seem to understand that. – tryst with freedom Apr 25 '22 at 21:50
  • @Buraian $$\frac {\partial \Delta t} {\partial t} $$ part of nabla means that time flow rate doesn't fluctuate as time passes, constant in time. – Agnius Vasiliauskas Apr 26 '22 at 05:00