# Is the concept of (Newtonian) forces part of General Relativity?

Let me clarify my question by explaining my thought-process to you:

We all know that in order to explain gravity, Newton introduced the concept of a gravitational force to explain the phenomenon of masses being attracted to other masses. Of course there are also other forces not related to gravity, i.e. when you tug a car you exert a force on it too, centripetal force etc.

Einstein however explained the phenomenon of masses being attracted to other masses by saying that naturally, everything follows a geodesic in spacetime. And because masses curve spacetime, the geodesic of both objects appear curved to other observers, giving rise to the phenomenon of them attracting each other.

Bottom line is: Einstein made the need to introduce the concept of a gravitational force obsolete because geodesics explain it all by themselves.

Does General Relativity also make the introduction of other forces (i.e. Coriolis-, centripetal- and normal force) obsolete by explaining its related phenomena through curved spacetime as well?

I hope the question is posed unambiguously and understandably. Here is my approach:

Taking the centripetal force as an example: If I sit inside a carousel, the rotational energy changes "my metric tensor", thus curving spacetime around me resulting in me having a circular motion. Therefore I can describe my motion solely by my changed metric tensor and curved spacetime outside the Newtonian framework without needing to talk about a centripetal force.

My question is not whether the above phenomena can be explained by GR, which I know they can. It is whether GR completely overturns the idea/concept of a force. What makes me question this is the fact that I keep hearing people say:

"An object follows a geodesic in spacetime unless a force acts on it, knocking it off the geodesic. An example of that is you currently sitting on your chair."

So there are forces in GR?! :D

Force, in its Newtonian sense, is an integral part of general relativity. General relativity attributes the effects of gravitation and inertia to a dynamic 3D metric and a "dynamic" time lapse. But all other forces remain as they were. In fact we can say that it's force, momentum, and energy – not mass – that cause curvature and are affected by curvature.

This is clear from the Einstein equations: $$\pmb{G} = \kappa\pmb{T} \equiv \kappa\begin{pmatrix} \text{energy density}&\text{energy flux}\\ \text{momentum density}&\text{stress} \end{pmatrix}$$ where stress is basically the classical force/area and includes all standard contact forces such as pressure, friction, tension, viscosity, and so on.

This is even clearer if we write the Einstein equations in 3+1 form, choosing a particular but arbitrary frame in spacetime. Such a frame is chosen in two steps. First we choose a slicing of spacetime into 3D spacelike hypersurfaces, which can be labelled by an arbitrary parameter $$t$$. Then we (arbitrarily) identify points among such hypersurfaces – we are basically saying "I decree that this point on the hypersurface $$t'$$ is the same as this point on the hypersurface $$t''$$", thus arbitrarily decreeing what is at rest and what isn't. Such a choice is represented by a so-called lapse function $$\lambda$$, which relates proper time with our arbitrary labelling $$t$$, and a so-called shift vector $$\pmb{v}$$, which connects points "at rest" across the 3D spacelike hypersurfaces. For details and much better explanations see for example York (1979), Smarr & York (1978), Smarr & al. (1980), Marsden & Hughes (1994) § 2.4, Misner & al (1973). Note that this notion of frame is much freer than in Newtonian mechanics, where frames are traditionally chosen with respect to universal time and an unchangeable spatial metric.

The Einstein equations in 3+1 form can be found for example in Gourgoulhon (2012) chs 4–6, Alcubierre (2008) ch. 2, Rezzolla & Zanotti (2013) ch. 7, or Wilson & Mathews (2007) ch. 1: \begin{align} \partial_t \pmb{g} &= -2 \pmb{K} \pmb{g} \tag{1}\label{1} \\ \partial_t \pmb{K} &= \pmb{K}\ \mathrm{tr}\pmb{K} - \pmb{K}^2 + \pmb{R} + \frac{\kappa}{2}\ (\mathrm{tr}\pmb{\tau} - \epsilon) - \kappa\ \pmb{\tau} \tag{2}\label{2} \\ \partial_t \epsilon &= \epsilon\ \mathrm{tr}\pmb{K} - \nabla\cdot \pmb{p} + \mathrm{tr}(\pmb{K}\pmb{\tau}) \tag{3}\label{3} \\ \partial_t\pmb{p} &= \pmb{p}\ \mathrm{tr}\pmb{K} - \nabla\cdot\pmb{\tau} \tag{4}\label{4} \end{align} In these equations we have chosen a free-falling frame of reference (unit lapse function $$\lambda$$ and zero shift vector $$\pmb{v}$$).

• $$\partial_t$$ (really a Lie derivative) is the time derivative with respect to a place fixed in this frame;

• $$\pmb{g}$$ is the 3-metric of a 3D spacelike slice, with Ricci curvature $$\pmb{R}$$ and covariant 3-derivative $$\nabla$$;

• $$\pmb{K}$$ is the extrinsic curvature of such a slice within the 4D spacetime (note that it is independent of $$\pmb{g}$$);

• $$\pmb{p}$$ is 3-momentum density, which is equivalent to an energy flux in general relativity;

• $$\epsilon$$ is energy density;

and finally

• $$\pmb{\tau}$$ is the ordinary stress, that is, any kind of contact force per area.

Equation $$\eqref{2}$$ shows how 3D space changes its curvature with time: this change depends on the distribution of energy density and of contact forces, $$\pmb{\tau}$$, within the slice. Equation $$\eqref{4}$$ is basically the classical Newtonian equation relating the change of momentum to force; one can show that the term $$\pmb{p}\ \mathrm{tr}\pmb{K}$$ corresponds to convection of momentum, also present in Newtonian continuum mechanics (see eg Marsden & Hughes 1994: $$\pmb{K}$$ is related to the symmetrized gradient of the velocity $$\pmb{u}$$ of a matter element, $$\tfrac{1}{2}(\nabla\pmb{u} +\nabla\pmb{u}^{\intercal})$$). "Fictitious forces" such as gravitation or Coriolis appear in this equation when we choose a different frame, represented by a non-unit lapse function $$\lambda$$ and a non-zero shift vector $$\pmb{v}$$. The general equation in an arbitrary frame is (warning, I may have some signs wrong) $$(\partial_t - \mathrm{L}_{\pmb{v}})\pmb{p} = \lambda\ \pmb{p}\ \mathrm{tr}\pmb{K} - \nabla\cdot(\lambda \pmb{\tau}) - \epsilon\ \nabla\lambda$$ where $$\mathrm{L}$$ is the Lie derivative. See for example eqn (6.20) in Gourgoulhon (2012). The term $$\epsilon\ \nabla\lambda$$ is related to the gravitational force in the Newtonian form $$\rho\ \nabla\Phi$$, where $$\rho$$ is mass density. Gourgoulhon (2012), ch. 6, again gives a nice dicussion of the Newtonian limit of the equations above.

So general relativity says that the relation between force and rate of change of momentum is affected by curvature (the term with $$\mathrm{tr}\pmb{K}$$) and by our arbitrary choice of reference frame (the terms with $$\lambda$$ and $$\pmb{v}$$). Our reference frame actually affects all equations above (see cited references).

Regarding geodesic motion: most matter actually does not follow geodesic motion. The parts of our bodies, the parts of the computer or phone in front of us, water in a pipe, the oceans, the plasma in stars, do not follow geodesic trajectories. This is clear from the equations above (to which we must add an equation of conservation of matter). In fact, geodesic motion is quite a tricky concept in general relativity. We can't consider pointlike particles – "Dirac deltas of rest mass", so to speak – because they would generate singularities: general relativity is at heart a field or continuum theory, not a particle theory. Matter must always be considered as "smeared out". If some smeared-out matter can be completely surrounded by a world-tube with small cross section, then such a world-tube is approximately described by a geodesic. But small matter elements within such world-tube are typically not following geodesics. On this problem see for example the review by Infeld & Schild (1949) and Geroch & Soo Jang (1975).

The role of force in general relativity is often a confusing topic. In my opinion part of the problem is that most physics curricula don't teach Newtonian continuum mechanics anymore – yet general relativity is based on it, as clear from the important role that classical stress $$\pmb{\tau}$$ and its 4D generalization, the stress-energy-momentum tensor $$\pmb{T}$$, play in it. I recommend the text by Marsden & Hughes (1994), which gives an overview of Newtonian continuum mechanics with many connections with Lorentzian relativity, and also the beautiful article by Eckart (1940). The brilliant text by Frankel (1979) also gives a deeper understanding of the transition "from force to curvature".

# References

• Alcubierre (2008): Introduction to 3+1 Numerical Relativity (Oxford).

• Eckart (1940): The thermodynamics of irreversible processes. III. Relativistic theory of the simple fluid. Phys. Rev. 58, 919.

• Frankel (1979): Gravitational Curvature: An Introduction to Einstein's Theory (Freeman).

• Geroch, Soo Jang (1975): Motion of a body in general relativity. J. Math. Phys. 16, 65.

• Gourgoulhon (2003): 3+1 Formalism in General Relativity: Bases of Numerical Relativity (Springer).

• Infeld, Schild (1949): On the motion of test particles in general relativity. Rev. Mod. Phys. 21, 408.

• Marsden, Hughes (1994): Mathematical Foundations of Elasticity (Dover).

• Misner, Thorne, Wheeler (1973): Gravitation (Freeman).

• Rezzolla, Zanotti (2013): Relativistic Hydrodynamics (Oxford).

• Smarr, Taubes, Wilson (1980): General Relativistic Hydrodynamics: The Comoving, Eulerian, and Velocity Potential Formalisms. In Tipler (ed.): Essays in General Relativity: A Festschrift for Abraham Taub (Academic Press), 157.

• Smarr, York (1978): Kinematical conditions in the construction of spacetime. Phys. Rev. D 17, 2529.

• Wilson, Mathews (2007): Relativistic Numerical Hydrodynamics (Cambridge).

• York (1979): Kinematics and dynamics of general relativity. In Smarr (ed.): Sources of Gravitational Radiation (Cambridge), 83.

• +1 for the relevance of continuum mechanics on relativity. – Claudio Saspinski Oct 11 '20 at 17:53
• Thank you @ClaudioSaspinski! Nice to meet other physicists who share this point of view. – pglpm Oct 11 '20 at 18:07

Does General Relavity also make the introduction of other forces (i.e. coriolis- , centripetal- and normal force) obsolete by explaining its related phenomena through curved spacetime as well?

You're mixing together unlike things. A normal force is not a fictitious force. A coriolis force is a fictitious force, and is described that way both by Newtonian mechanics and by GR. GR also describes gravitational forces as fictitious forces. A centripetal force is simply one that happens to point in a certain direction (toward the center of the circle). It's not a different type of force.

everything follows a geodesic in spacetime.

No, only free-falling objects follow geodesics.

Taking the centripetal force as an example: If I sit inside a carousel, the rotational energy changes "my metric tensor"

No, the metric just is what it is. Your rotational energy is not enough to have any significant effect on the metric. The metric doesn't depend on your state of motion.

thus curving spacetime around me resulting in me having a circular motion.

No, you have circular motion because of frictional and normal forces exerted on you by the carousel.

Therefore I can describe my motion solely by my changed metric tensor and curved spacetime outside the newtonian framework without needing to talk about a centripetal force.

No, this is not true.

This is a comment:

Searching through the literature there are several derivations of Newtonian gravity from the General Relativity expressions. For small masses and velocities (flat space) Newtons equations are recovered. This is an example of the mathematics.

Where General Relativity has to be used forces have no specific meaning. The GR equations have to be solved.

In Newtonian physics, every force can be described by a (three-)vector, including fictitious forces in accelerating coordinates.

In special relativity, electromagnetic and normal forces can be described by (four-)vectors, but fictitious forces can't be. If you write special relativity in a manifestly covariant form with a metric tensor, then fictitious forces appear as changes in the metric tensor. You could say that there are two kinds of force in special relativity, vector and tensor.

General relativity adds the gravitational force, which turns out to be a tensor force.

I think that the situation as a whole is not much different from the situation in Newtonian physics, except that there are two categories of force instead of one.

I think that it is the opposite. In the framework of GR the concept of force keeps necessary and it is clearer than in Newtonian mechanics.

It is necessary to postulate a gravitational force in Newtonian mechanics, otherwise accelerated movements of planets and apples would violate the second law. And a typical device to measure force as a spring, would measure a force without acceleration for objects deflecting it due its (object) weight.

In GR, it is "downgraded" to a fictitious force. But we are still doing a real force upwards when hanging an object for example. And it is real because it can be measured by a load cell or a spring. On the other hand, we can not measure that way a body in free fall.

I get the impression that when people ask about the concept of force they assume that there a single universal definition of force. However, that is not the case.

In the history of physics it is quite common for a word to be radically redefined. Example: physics used to have 'Caloric theory', but over time it became clear that Caloric theory had to be abandoned. The word 'calorie' was not discarded, but in terms of the new theory (the dynamic theory of heat) the definition of the word calorie was changed profoundly.

Something similar has happened in the case of force

Let me describe a modern definition of force:

The electrostatic interaction, for example, has an associated potential. That is, when charged particles move closer to each other or away from each other there is a corresponding change of potential energy. In other words: the interaction between charged particles is such that there is a gradient in the potential energy. You have the option of attributing the motion along the gradient of the potential as due to a force. The magnitude of that force is given by the derivative of the potential with respect to the distance between the particles.

This modern definition of force replaces the newtonian definition of force.

Newtonian definition of force:
Force is that which causes acceleration with respect to an inertial coordinate system.

Clearly, the newtonian definition of force is useless in the context of GR. The example that everybody gives: the orbital motion of celestial bodies. Take for example Halley's comet. The center of mass of Halley's comet is in inertial motion. So, sure: in terms of the newtonian definition of force Halley's comet is not subject to a force.

In terms of the modern definition of force the assessment is:
Is there such a thing as gravitational potential energy? Yeah, there is. The magnitude of the force that you attribute to the gravitational interaction is given by the derivative of the potential with respect to the distance between the masses.

As Halley's comet moves from aphelion to perihelion gravitational potential energy is converted to kinetic energy. Then after the perihelion Halley's comet climbs up the potential gradient; kinetic energy is converted to potential energy.

Asking the right question is crucially important.

The question 'is Gravity a force?' is the wrong question to ask; the point is moot.

The right question to ask is:
Is there interconversion between potential energy and kinetic energy?