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## Main Question or Discussion Point

I am having trouble connecting some of the differential geometry in gr to what is actually measurable in the real world.

As far as i understand we can measure physical quantities in terms of coordinates [itex] x^\mu [/itex] on the tangent space of a 4-dim Pseudo-Riemannian manifold M. So lets say we are in the tangent space at some point p on M and observe a free falling particle on a geodesic passing trough that point and we measure some acceleration [itex] \ddot{x^\mu} [/itex]. Since it is in free fall the geodesic equation must be satisfied

[itex] (\nabla_{\dot x} \dot{x})^\mu = \ddot{x^\mu}+\Gamma^{\mu}_{\sigma\delta}\dot{x}^\sigma\dot{x}^\delta = 0[/itex].

Is it correct to say that we see the coordinate acceleration [itex] \ddot x^\mu [/itex] because the velocity vector is trying to compensate for the change of the basis vectors of [itex]T_{p}M[/itex] (or equivalently change of the metric ), by changing it's coordinates with respect to these in order to keep the intrinsic length of the vector fixed (hence covariant derivative of it equals zero)?

Would that mean that some hyper dimensional being, that could see our 4-dim space-time as embedded in some R^n, would see the basis vectors of the tangent spaces (in an appropriate chart) change in size just as i can envision them changing from point to point on the 2-sphere. So he would basically see a tiny human with a ruler in his hand measuring stuff that is actually of constant size with a ruler of variable length (coordinates of ruler stay the same but basis vector changes) and ascribing the change to said quantities rather than to his ruler changing?

If so I could finally understand the redshift of distant galaxies due to expansion of universe in the sense that the wavelength is some extrinsic quantity that stays constant, but the basis vectors are shrinking everywhere making our rulers/coordinates smaller/worth less, if seen from the outside, hence leading us to measure an increased wavelength.

So again to summarize you have something that wants to stay constant, but space-time is contracting/expanding beneath it's feet, so to us it has to look like it is changing basically because we change with space-time.

This way of explaining things makes sense to me but i've never been really able to read this out of a textbook (maybe I don't read carefully enough hehe).

Could you please point out any flaws in my point of view?

As far as i understand we can measure physical quantities in terms of coordinates [itex] x^\mu [/itex] on the tangent space of a 4-dim Pseudo-Riemannian manifold M. So lets say we are in the tangent space at some point p on M and observe a free falling particle on a geodesic passing trough that point and we measure some acceleration [itex] \ddot{x^\mu} [/itex]. Since it is in free fall the geodesic equation must be satisfied

[itex] (\nabla_{\dot x} \dot{x})^\mu = \ddot{x^\mu}+\Gamma^{\mu}_{\sigma\delta}\dot{x}^\sigma\dot{x}^\delta = 0[/itex].

Is it correct to say that we see the coordinate acceleration [itex] \ddot x^\mu [/itex] because the velocity vector is trying to compensate for the change of the basis vectors of [itex]T_{p}M[/itex] (or equivalently change of the metric ), by changing it's coordinates with respect to these in order to keep the intrinsic length of the vector fixed (hence covariant derivative of it equals zero)?

Would that mean that some hyper dimensional being, that could see our 4-dim space-time as embedded in some R^n, would see the basis vectors of the tangent spaces (in an appropriate chart) change in size just as i can envision them changing from point to point on the 2-sphere. So he would basically see a tiny human with a ruler in his hand measuring stuff that is actually of constant size with a ruler of variable length (coordinates of ruler stay the same but basis vector changes) and ascribing the change to said quantities rather than to his ruler changing?

If so I could finally understand the redshift of distant galaxies due to expansion of universe in the sense that the wavelength is some extrinsic quantity that stays constant, but the basis vectors are shrinking everywhere making our rulers/coordinates smaller/worth less, if seen from the outside, hence leading us to measure an increased wavelength.

So again to summarize you have something that wants to stay constant, but space-time is contracting/expanding beneath it's feet, so to us it has to look like it is changing basically because we change with space-time.

This way of explaining things makes sense to me but i've never been really able to read this out of a textbook (maybe I don't read carefully enough hehe).

Could you please point out any flaws in my point of view?