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As I understand, the Theory of General Relativity says that the force of gravity doesn't exist. The planets move in elliptical orbits because the spacetime is curved by the mass of the sun, not because there is any force similar to Coulomb force between electrostatic charges.

So, why to unify gravity with electromagnetic, week and strong nuclear forces if it doesn't exist? Even I would ask, why everybody says that there are 4 fundamental forces in Nature instead of 3 forces in a 4 dimensional manifold? What is the role of the graviton as a boson gauge if there isn't any force to mediate?

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    $\begingroup$ Related: physics.stackexchange.com/q/6980/2451, physics.stackexchange.com/q/61899/2451 and links therein. $\endgroup$ – Qmechanic Oct 18 '17 at 18:18
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    $\begingroup$ The statement "gravity is not a force" is not the same statement as "gravity doesn't exist". The statement "The force of gravity does not exist" is technically true, but only because gravity isn't a force - it definitely exists. $\endgroup$ – tparker Oct 18 '17 at 23:25
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    $\begingroup$ Try replacing force with interaction? $\endgroup$ – Alfred Centauri Oct 19 '17 at 0:13
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At the beginnings of the XX century, Kaluza made an interesting exercise by writing down a 5D version of General Relativity, but it turned out to be really complicated and hard to interpret. So he made a simplifying assumption: He assumed the metric does not depend on the 4th spatial coordinate. What he obtained was the metric resulted in the normal 4D metric that we are familiar with, but he also obtained a vector field, and a scalar that measured dilation in the 4th spatial dimension.

When he analyzed the dynamical equations governing the vector field, he discovered that the equation was the same governing the electromagnetic potential 4-vector, once he equated linear momentum on the 4th dimension to electric charge. He also had to assume the scalar was constant, as he wasn't able to make a connection with physical reality

Attempts to incorporate this 'fields as geometry' view with the knowledge of QFT and the standard model was taken by string theorists with variable degrees of success. Einstein continued working on a classical approach toward unification until he died, but nowadays no one pursues this program anymore due to the difficulty to incorporate quantum dynamics into them

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The force is real, curvature doesn´t make a difference. Saying that it is caused by curvature is wrong, because the curving has to be done by a force that is equal in power.

Unifying? Well, heat is electromagnetic radiation in units W/m^2, and gravity is a force in units Nm. To compare the force and the heat like in a heat engine with the first law, set gravity to act as stress/pressure/thermal resistance, in units Nm^2. Then heat emission at earth surface is equal to the source power density of a gravitational force as resistance, $4g^2$. Surface temperature at 287K is equal to heat flow of 383W/m^2, and a thermal resistance of $383Nm^2=4g^2$. First law of thermodynamics with irradiation by the sun(TSI, $1360.8W/m^2$) as dU:

$TSI(dU)-4g^2(W)=4Q=4*244.5W/m^2=$ effective emission at 256K

So, gravity and heat emission from effective temperature, as heat and work in the first law. That is unified enough for me.

I saw something about charge in another answer. It is interesting that solar heating and gravity is equal in the equation for the electric field inside a hollow sphere with a ball at the center, $TSI/(4/3)=4/3*8g^2$ The relation where volume holds the sum of sources for the surface flux imitates real relationships accurately. This is Gauss law for gravity and the shell theorem. It fits perfectly on earth, and the idea of charges, potentials, in the heat flow, makes me think electricity seems to hold the answers to gravity and hot bodies in space.

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Whatever anyone say, gravity is a force. The curving of space is GR model that accurately quantifies the effects. GR may be meaning that force is caused via curving of space.

But GR is a very unique model which almost gives impression of gravity not being a force. So, it is possible that the unification might have been made somewhat more difficult due to GR, can not say for sure.

But as you asked - Why to unify gravity with the other forces if gravity isn't really a force?

One can also ask - How can anyone unify forces with something that is not a force? So, the unification has been ever evading.

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