Whatever nudge rpenner has given anyone, it pales into insignificance with the bump he has given you.

This thread is much ado about nothing and you were exposed at the beginning.

Let me again sum of the gist of this thread:

Everything will travel in a straight line unless acted on by contrary forces [Newtonian] or is influenced by the geometry of spacetime. [GR]

When you finally accept that oh so obviously correct answer, then you may salvage whatever reputation you have left.

Goodnight, I need my beauty sleep.

Paddoboy -- perhaps it is your lack of sleep but you have been terrible at summarizing GR and terrible at supporting your points in persuasive manner. It doesn't matter if you are conveying your best understanding of this subject when you don't have more than a gloss of it and sometimes are wrong. It's not my job to correct every mistaken claim, but when people shroud themselves in ego and assumed positions of righteousness, it makes my job harder. The God may be more interested in tweaking your nose than arguing like a seeker of knowledge. So you need to be a responsible adult and support your points better without having to post multiple times per hour.

The issue has always been one of definitions. You can't argue with definitions. You can't switch back and forth between different definitions of straight line and force in the same discussion because you won't be talking about the same thing. That's equivocation, which should be called out as a foul.

*Everything* will travel in straight lines unless acted on by a net non-zero force. That's very nearly the definition of force in terms of straight lines. So what is the definition of straight line? The God hasn't come close to engaging this basic question, so probably is a troll. Farsight won't even engage with the subject even when presented with Einstein's formal definitions, which is pretty damn ridiculous given his

*modus operandi* of running to Einstein's papers as the purported sole source of authority.

• Newton assumed absolute Euclidean space and absolute Euclidean time so his straight lines are those of Cartesian coordinates where $$x''(t) = 0, y''(t) =0, z''(t) = 0$$. Newton was able to unify terrestrial and celestial motions by assuming gravity was a force. Later people would describe electromagnetism as a force and wonder why inertial mass and gravitational charge were one and the same.

• Einstein, in Special Relativity, rejected separate space and time and connected them into a structure of a new sort of geometry where the straight lines in Cartesian coordinates are $$x''(\lambda) = 0, y''(\lambda) =0, z''(\lambda) = 0, t''(\lambda) =0$$ but the ones available to matter those where : $$(ct')^2 - (x')^2 - (y')^2 - (z')^2 > 0$$. These two combine to give basic agreement on what are straight lines with Newton, but now the coordinate velocity of material particles are limited to be below the speed of light in vacuum. We get the twin paradox in Special Relativity because a straight line is the

**longest** path available to material particles, for the metric of this geometry is basically the same thing as proper time. But Special Relativity could not be tweaked to give a satisfactory theory of gravity.

• Einstein, in General Relativity, went to 19th century descriptions of manifolds and tweaked them to have the same local geometry as special relativity. Now Cartesian coordinates are much less special because it is geometry that dictates what a straight line is. The tool that connects coordinates and geometry is the symmetric metric, $$g_{\mu\nu}$$, but unlike earlier manifolds, now it is not positive-definite. The definition of a straight line is now

$$\forall \mu \in \{ 0,1,2,3 \} \; \sum \limits_{\nu \in \{ 0,1,2,3 \} } g_{\mu\nu} x''^{\nu}(\lambda) + \sum \limits_{\alpha, \beta \in \{ 0,1,2,3 \} } \frac{1}{2} \frac{ \partial g_{\beta\mu} }{ \partial x^{\alpha} } x'^{\alpha}(\lambda) x'^{\beta}(\lambda) + \frac{1}{2} \frac{ \partial g_{\mu\alpha} }{ \partial x^{\beta} } x'^{\alpha}(\lambda) x'^{\beta}(\lambda) - \frac{1}{2} \frac{ \partial g_{\alpha\beta} }{ \partial x^{\mu} } x'^{\alpha}(\lambda) x'^{\beta}(\lambda) = 0$$

which is painful in coordinate-based analysis, but in geometric language can be as simple as $$\mathbf{\nabla _ u u} = 0$$ which says a straight line is compatible with parallel transport on a curved manifold of its own tangent vector. Likewise the straight lines available to material particles are those where $$\sum \limits_{\mu, \nu \in \left\{ 0,1,2,3 \right\} } g_{\mu\nu} x'^{\mu} (\lambda) x'^{\nu} (\lambda) $$ has the correct sign (which is a matter of convention for defining the metric). Now, in the math and philosophy of general relativity, the phenomena of gravity is not described as a force, but as how the content of the universe shapes its geometry.