In the attempts to quantify General Relativity, there are significant conceptual problems in the way time is conceived as a dynamic metric of space-time. The notion of time is embedded to the dynamic space-time geometry, and in quantifying that geometry, notion of time becomes self-referential to the dynamics of the geometry that describes time in the first place.

Click here for some commentary on the issue. Notice how typical attempts to quantize gravity involve quantizing general relativistic space-time itself. Notice how that leads into ill-defined background for quantum fluctuations. If it is space-time that fluctuates, what is it fluctuating in relation to?

Notice also the comments about problematic conceptualization of time;

*“Since time is essentially a geometrical concept [in General Relativity], its definition must be in terms of the metric. But the metric is also the dynamical variable, so the flow of time becomes intertwined with the flow of the dynamics of the system”*

If we still step back to Special Relativity for a bit, it's worth noting that usually the space-time is described with the metric;

$$ds^2= dx^2 + dy^2 + dz^2 - (cdt)^2$$

The time component $t$ is taken as a coordinate axis, and its signature means the interval $s$ is exactly 0 in the case that $dx^2 + dy^2 + dz^2 = (cdt)^2$.

Meaning, anything moving with the speed of light is described as having 0 length interval. It takes the same coordinates when it is at earth, as when it is in Alpha Centauri, in terms of its own coordinate system. Or to be more accurate, such coordinate system is ill-defined; "light" is conceptually time-dilated to 0 "it doesn't see time", and thus it doesn't see "t"-coordinate, in so far that one wishes to describe time as geometry. On the flip-side of the same coin, the spatial distance between events is also exactly 0, or to say it another way, there's no way to describe time evolution from the perspective of light.

Note that this convention is instantly thrown out of the window when one wishes to move to General Relativity. (See the quote from "Gravitation")

However, typical attempts to quantify gravity still tend to conceptualize time as geometrical axis of the coordinate system where the data is displayed.

Despite these kinds of conventional difficulties, Richard's analysis at hand reproduces both quantum mechanical and relativistic relationships from the same exact underlying framework, and in doing so it does in fact give a quantified representation of relativistic gravity, without employing the concept of space-time at all. In other words, while the presented framework is not really a theory about reality per se, it does effectively give a fully general representation of any and each possible unifying theory, or so-called "Theory of everything".

Naturally, unifying both relationships under the same logical framework begs the question, what exactly is the critical difference between Richard's framework, and the conventional form of these theories?

Many definitions picked up from the analysis correspond almost identically to concepts also defined in conventional physics - e.g., objects, "energy", "momentum", "rest mass" (p. 56-58) - there is one very crucial difference with the way time is conceptualized. Think carefully of the following;

The parameter $t$, or "time", is not defined as a coordinate axis. It was defined explicitly as an ordering parameter; elements associated with the same $t$ are taken to belong to the same "circumstance" (it is meant as an ordering parameter). And if you follow the development of general rules (p. 14-40), you can see that under this notation, elements must be seen as obeying contact interactions. Meaning, only the elements that come to occupy the same position at the same "time" can interact. Notice how that statement about "time" is not consistent with the idea that clocks measure "time"; what clocks measure is a different concept and great care must be taken to not tacitly confuse the two at any point of an analysis.

However, $\tau$, or tau, was defined as a

**coordinate axis**, originally having nothing to do with time. It was named as tau at the get-go because in the derivation of Special Relativity, $\tau$ displacement turns out to correspond to exactly what any natural clock must measure; closely related to the relativistic concept of proper time which is typically symbolized with $\tau$. It is important to understand the difference though; in Special Relativity, $\tau$ is NOT a coordinate axis! Under the specific paradigm that we call Special Relativity, it cannot be generalized as a coordinate axis. The associated definitions of the entire framework need to be built accordingly.

Remember, in the analysis, $\tau$ was an "imaginary" coordinate value at the get-go; object's position in $\tau$ yields no impact in the evaluation of the final probabilities; its value is to be seen as completely unknown. On the other hand, object's momentum along $\tau$ corresponds to its mass (Eq. 3.22), which is essentially a fixed quantity, and thus treated as completely known.

Which simply means that $\tau$ position cannot play a role in the contact interactions of the elements. You may conceptualize it as if every object is infinitely long along $\tau$, if you wish. Either way, projecting $\tau$ position out from object interactions is the notion which is typically amiss in most attempts to describe relativity with euclidean geometries, and without it you tend to become unable to define interactions meaningfully.

Note how this corresponds exactly to the fact that the fundamental equation we are working with, is a wave equation. Since it is a wave equation, it is also governed by uncertainty principle from bottom-up.

Meaning mass $m$ and $\tau$ must be related with $\sigma_m \sigma_{\tau} \geq \frac{\hbar}{2}$. Since the momentum along $\tau$ (defined as mass) is completely known, the $\tau$ position must be completely unknown.

Note that under conventional description of Relativity, while $t$ is a coordinate axis against which all data is plotted, there is no such thing as clock that measures $t$; each clock measures its own proper time, $\tau$. We just cast $t$ from $\tau$ measurement. In so far that we are willing to approximate our clocks as stationary (which they never are), and approximate the effects of gravity and its unpredictable fluctuations, that casting is rather trivial. But when you involve gravity, you involve curving and wiggling world lines to all the elements, including the time reference devices, and the relationship between $\tau$ and the coordinate axis you wish to plot your data with (t) becomes much harder to handle. It seems quite rational to investigate a paradigm where $\tau$ is in fact by its very definition a coordinate axis.

I won't repeat the derivation in the book, but please post a comment if anything in the derivation seems unclear, so it can be clarified by the author.

Few comments are in order though. Note that the final result of the deduction (Eq. 4.23) is not completely identical to Schwarzschild solution; it contains an additional term, whose impact is extremely small. This is rather interesting, because it implies a possibility for experimental verification. On the other hand, it also may be simply an error by the author. I think this part would require that enough experienced people would walk through the derivation and see if they can find errors. With so many people looking for a way to describe quantum gravity, I would think there are interested parties out there.

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