# Inveiling the breakdown of Euclidean geometry in linearized general relativity

 Type classification: this is an essay resource.
A planet consists of a large but very thin ultra-dense disk that generates earthlike gravity.The planet occupies the y = 0 plane. The metric is such that inhabitants on each side would perceive their world to be bowl shaped. Curve (a) describes the path of light. Curve (b) is a straight line, but its equation is rendered complicated due to the fact that (x,y) are not cartesian coordinates.
Concept map for deriving linearized relativity in a way that unveils the breakdown of Euclidean geometry.

This page is devoted to the discussion of a "derivation" of linearized general relativity that can be seen by reading either of these two pdf files:

#### We begin with a false assumption

I have tagged my own essay as "fringe science" partly because it is based on an equation that cannot be true because it involves variables that do not exist:[1]

(1)   ${\displaystyle x=x(x^{*},y^{*},z^{*},t^{*})}$,    ${\displaystyle y=y(x^{*},y^{*},z^{*},t^{*})}$,    ${\displaystyle z=z(x^{*},y^{*},z^{*},t^{*})}$,    ${\displaystyle t=t(x^{*},y^{*},z^{*},t^{*})}$ ,

where ${\displaystyle (x^{*},y^{*},z^{*},t^{*})}$ are the spacetime variables a Minkowskian space that is 'flat' in that it lacks Riemannian curvature. While this seems like a legal "change of variables", or transformation between two coordinate systems, the claim that ${\displaystyle (x^{*},y^{*},z^{*},t^{*})}$ exist as Minkowskian variables is false, as this "derivation" ultimately establishes. If space-time is curved, there is no coordinate system with the 'flat' Minkowskian metric. In other words, we take equation (1) as a starting point for the construction of a theory that eventually forces us to abandon it. It is understood that these variables do exist locally, and that the differential

(2)   ${\displaystyle dx^{*2}+dy^{*2}+dz^{*2}-dt^{*2}}$

is an invariant "distance" between two nearby events in space-time.

#### Postulate a simple field equation

I am not an expert in General relativity, and my only claim to fame in this field is that I read that I read about 10% of Gravitation and understood most of what I read. One thing I did not fully understand is why a scalar theory of gravity fails. Instead of struggling to grasp what was apparently commonly understood, I decided instead to accept it as fact and proceed to a rank 2 tensor theory that was modeled after the potential version of electrodynamics in the Lorentz gauge. Instead of Maxwell's,

(3)   ${\displaystyle \Box {\stackrel {\leftrightarrow }{\Phi }}=\left[{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right]{\stackrel {\leftrightarrow }{\Phi }}\approx -16\pi {\stackrel {\leftrightarrow }{\mathrm {T} }}}$

where T is the Stress–energy tensor, and in the non-relativistic Newtonian theory of gravity, and the only non-zero element of the potential is the time-time component, which equals −4V where V is the Newtonian gravitational potential, i.e. , −∇V=g is the local gravitational acceleration.

#### Why was an approximate equation postulated?

${\displaystyle \nabla ^{2}}$ = ∂2/∂x2+∂2/∂y2+∂2/∂z2 is an operator that does not have the exact symmetry under the Lorentz transformation because such a symmetry is associated only with the ficticious (or local) spacetime variables are ${\displaystyle (x^{*},y^{*},z^{*},t^{*})}$. To achieve approximate validity we assume that the gravity is weak enough that,

${\displaystyle x\approx x^{*},\;\;y\approx y^{*},\;\;z\approx z^{*},\;\;t\approx t^{*}}$.

these approximations become exact in the limit of free space (i.e. no gravitational objects present). The decision to transform from the (ficticious) Minkowskian coordinates to these new (x,y,z,t) variables represents an attempt to reconcile three facts:

1. Energy and frequency are related by De Broglie's ${\displaystyle E=\hbar \omega }$
2. A photon loses frequency (energy) as it rises against a gravitational field.
3. The angular frequency ${\displaystyle \omega }$ is time invariant for a static gravitational field (e.g. for a spherical mass situated at the origin)

The latter statement can be expressed within the scope of classical mechanics of point particles as dH/dt = ∂H/∂t, or in terms of a wave equation, where if the coefficients are all time invariant, then solutions are of the form exp(i−ω0t). This does not prove that the gravitational redshift is incompatible with a 'flat' coordinate system, but merely that a theory in such a system would be a bit complicated. Consider an attempt to construct a theory of motion for a massive stationary sphere situated at the origin. In order to construct are realistic theory of motion and waves, it would be necessary for the Hamiltonian to be time dependent.

To avoid this complication, we construct our theory in slightly different coordinates. If ${\displaystyle \omega ^{*}}$ and math>\t^*[/itex] represent the actual frequency and time, then for a stationary observer then we have this relationship bewteen the evolution of a wave as expressed in the two coordiant systems:

${\displaystyle \omega dt=\omega ^{*}dt^{*}}$

This permits to observed frequency ${\displaystyle \omega ^{*}}$ to depend on position, while the variables are such that ${\displaystyle \omega }$ is a constant of motion (i.e. invariant with respect to t which does not exactly present time).

#### Critique of this "derivation"

I developed this theory after having become proficient in vector wave equations in theoretical plasma physics, where curvilinear coordinates are sometimes used to model the geometries of a tokamak or mirror confined plasma. Although these coordinate systems were exotic, they were always cast in a Euclidian space, and most models were non-relativistic. I always had the concept of (x,y,z,t) to fall back on as I contemplated the situation. How can you do physics without (x,y,z,t)? Even special relativity has (x,y,z,t). So I decided to construct a rank 2 tensor theory in a 'flat' (x,y,z,t), fully aware that I would have to adjust. The need for adjusted coordinates quickly emerged through the gravitational redshift, which for weak gravity is easily deduced using simple physical arguments. But there was no immediate breakdown of Euclidean geometry.

The derivation was never intended to be rigorous, and for that reason would not likely have been discovered until after General Relativity was already established. Nor is it likely that this derivation is likely to revolutionize General Relativity, which appears to the second most stable theory in the history of physics (second only to Newton's laws of motion)[2].

This discussion should be understandable to any student with a strong background in electrodynamics, Hamiltonian physics, and vector calculus that includes an understanding of how the gradient and Laplacian in spherical coordinates can be derived. The derivation is relatively straightforward only if the observed gravitational of light by the sun is used to derive the equations. An appendix shows that it seems to possible to actually predict this anomalous bending of light by gravity, but the argument is tricky.

## Footnotes and references

1. And also because nobody seems to like it.
2. Maxwell's theory of electromagnetism was almost immediately modified by quantum field theory, and quantum mechanics has endured fundamental modification almost continuously since the Bohr atom was proposed.