Precipitate Lorentz covariance of EM

 Unifying Electromagnetism and Gravity is the love of my youth and such loves are never forgotten. 

Three years ago I had the impression that I still were not ready to undertake the task, for I encountered many confusions and a myriad of possibilities, all of which seemed `beautiful'. 

What I tried then was to take EM and GR as inputs hence I worked with them without even altering any details. This led me to many fruitful observation and influenced my later works. Although I learned many things and found some regularities and relations, which were exciting at that time, a deaf voice kept telling me `this is not the thing'. 

I tried extra dimensions along the way, went to as many as 7 dimensions. I tried non-linear theories like Born-Infeld and others. Needless to say I reflected --if not worked-- on most of the approaches mentioned in Goenner's great review. 

Long story short, I tried, and contemplated everything, and yet the voice kept yelling. 

To be honest, I was so confused and disappointed that I decided to give it up, at least temporarily. In the meantime I kept experiencing with `theory crafting' and picked up many things about theories, their structure, their epistemology, the forms of the their vulnerabilities, `range of' paradoxes in a theory's foundations, etc. I did a whole lot of work in both physics and mathematics, most notably, the greatest discovery of my life, acceleration of light.

A pivotal strand of thought that I explored was to put Electromagnetism and General Relativity on equal ground, separately first. This exploration thought me so many things, most important of which I believe is something that I cannot express in words properly. Roughly speaking, this exploration taught me the utility boundaries of formal thought and delineated that amount of formal similarity which is `good'.

I think now the time is ripe for returning to and settling the work. If I cannot succeed this time, chances are I never will.

Putting EM and GR on equal ground is where I continue the strand. This guiding principles allows us to learn from both and be able to apply methods/facts, etc. from one to the other. 

We have two facts:

1. Four-dimensions only. Here Conservativism is for the best for the alternatives are all less plausible and failed. This means we have to limit ourselves to four-vectors, in particular four-potential. This is the GR part; where we apply a GR result to EM.
2. Time has to enter the potentials directly. Observe that, since instead of $$\phi=-\frac{GM}{\sqrt{r^2 - (ct)^2}},$$ we have $$\varphi=-\frac{GM}{r},$$ GR is still not `relativistic' properly! This is the EM part; where we apply an EM result to Gravity itself.

Considering these facts as a whole, we are led to our first conclusion:

Gravitational field g should become a four-vector:

$$g_\mu = \partial_\mu \phi,$$ 

where $$\phi=-\frac{GM}{\sqrt{r^2 - (ct)^2}}.$$