General Relativity, is gravity a force?

[AdkinsJr]

In general relativity, gravity is modeled as a curvature in spacetime. So the Earth moves in a 'straight' path in a curved space. At least, this is how it has been explained to me in the past. The sun isn't actually 'pulling' on the earth, the Earth is just moving around in the 'gravitational well' of the sun. So in this sense, there is no 'force' of gravity is there? It's just a dynamic of spacetime.

I don't understand how gravity can be one of the four forces of nature if it's just curved spacetime. How is gravity still a force in general relativity?

 

[cybertific]

gravitation is a force. space time curvature in general relativity is just a tools to explain in another way

 

[pliu123123]

in every era, human beings have different understanding of the nature.
may be we can treat gravitational force as the MOTHER of concept of curved space-time
or in literal way, curved spacetime is the "improvement" of the concept of gravitational force

at least curve spacetime concept can describe the happen of the invariance of the speed of light.but not in the concept of gravitational force,
of course curve spacetime concept requires more complex formulation

 

[Phrak]

As you guessed, gravity is not a force in general relativity.

This is a change from Newtonian physics. An object that falls toward a gravitating body in Newtonian physics is subjected to the force of gravity. In general relativity no force acts upon the object. It is said to be 'freely falling'.
Beware, there is more than one way the word 'force' is used in physics. The four fundamental forces include the strong force, the weak force, the electromagnetic force and gravitational force.

 

[haushofer]

gravitation is a force. space time curvature in general relativity is just a tools to explain in another way

How would you then define "acceleration due to a gravitational field" if the equivalence principle holds?

 

[xlines]

In general relativity, gravity is modeled as a curvature in spacetime. So the Earth moves in a 'straight' path in a curved space. At least, this is how it has been explained to me in the past. The sun isn't actually 'pulling' on the earth, the Earth is just moving around in the 'gravitational well' of the sun. So in this sense, there is no 'force' of gravity is there? It's just a dynamic of spacetime.

I don't understand how gravity can be one of the four forces of nature if it's just curved spacetime. How is gravity still a force in general relativity?

Well, I think that depends on how you define concept of force. I would say gravity is force. First mass can change momentum of another one by an act of gravity? Therefor gravity is force! I don't know why people disqualify gravity as a force. Maybe because we don't have field theory of gravity, but IMHO that argumentation is void until someone shows gravity can not be quantized. Until then, statements like "gravity isn't a force" is an ideological extrapolation.

 

[TcheQ]

The acceleration of a mass towards another body is a byproduct of the curvature of spacetime.
If people argue this, then they don't understand it. Which will be common since four dimensions is one dimension bigger than our brain evolved to naturally conceptualise.

Well, I think that depends on how you define concept of force.

the concept of a force is a simplification we use of a physical process so our meagre intellects can interpret it e.g. Newtonian physics is a simplification of Einsteinian mechanics

For simplicity of calculations, we conform to the standardized F=mg

g=F/m

What you are saying it that you think gravity should be changed to units of kg.m/s/s and not the m/s/s that everyone else uses. In which case we would call it a force, and not gravity.

 

[dx]

Gravity, experimentally, is neither a force nor curvature. Experimentally it is simply a phenomena. Now, when you ask a question like "is gravity a force", you must always pose that question within a conceptual framework ('theory') that describes this phenomena. Newton's original theory of gravity is one such framework, and in that theory gravity is a force. Einstein's general relativity is another theory that describes the phenomenon, and in that theory gravity is represented in the formalism as curvature of spacetime.

Theoretical notions like 'force' are not properties that exist in nature, but ideas in a conceptual framework that is used to describe nature. For example, Newtonian mechanics can be reformulated in terms of the so called 'principle of least action', and the description of motion in that theory does not involve the idea of 'force', even though the two descriptions are completely equivalent.

 

[A.T.]

GR did not remove the "force status" from the "force of gravity". It just moved it to the "inertial forces", saying that if you are observing a "force of gravity" then you are not in an inertial frame.

 

[Nabeshin]

The above two posts do a good job uncovering some of the ambiguity of the original question, so I won't talk about that. What I do want to do is go into the GR description of the situation a little further. Like dx says, this is merely the interpretation within one framework.

In GR, the equation of motion for a free particle (i.e. no forces. Note: Gravity does not even exist in GR. So it makes no sense to speak of it as a force.) is the following:
$$\frac{d^2 x^\alpha}{d \tau ^2} = - \Gamma^\alpha_{\beta \gamma} \frac{d x^\beta}{d \tau} \frac{ d x^\gamma}{d \tau}$$

In the case of a flat spacetime, this reduces to:
$$\frac{d^2 x^\alpha}{d \tau ^2} =0$$
Which is precisely Newton's 2nd law for a free particle.

However, if we imagine our particle to be confined to the surface of a sphere, say, then it does NOT reduce to Newton's 2nd law. In this case, there are no identifiable forces (if we look at this classically, we do not even observe gravity!), and yet there is a deviation from the Newtonian motion.

All that changes when we introduce mass into the picture is that the $$\Gamma^\alpha_{\beta \gamma}$$ change in a predictable way.

No mention of the word gravity is ever needed.

This is the way I see it from GR, at least.

 

[TcheQ]

Good vid. 33:30 onwards talks a bit about this, and up to around 40:00, specifically the point regarding gravity and it's affect on light. (i.e if gravity were a force, it would not cause light to bend since light has zero mass)

 

[Altabeh]

The above two posts do a good job uncovering some of the ambiguity of the original question, so I won't talk about that. What I do want to do is go into the GR description of the situation a little further. Like dx says, this is merely the interpretation within one framework.

In GR, the equation of motion for a free particle (i.e. no forces. Note: Gravity does not even exist in GR. So it makes no sense to speak of it as a force.) is the following:
$$\frac{d^2 x^\alpha}{d \tau ^2} = - \Gamma^\alpha_{\beta \gamma} \frac{d x^\beta}{d \tau} \frac{ d x^\gamma}{d \tau}$$

In the case of a flat spacetime, this reduces to:
$$\frac{d^2 x^\alpha}{d \tau ^2} =0$$
Which is precisely Newton's 2nd law for a free particle.

I'm sorry, but what happens to the spacetime to be flat? The gravitational field exerts a force on both particles and the fabric of spacetime and if it is not present, the spacetime reduces to a flat spacetime so all particles move along straight lines! If it is present, then the particles would have proper accelerations due to the force acting on them!

(i.e if gravity were a force, it would not cause light to bend since light has zero mass)

The gravity does not affect photons directly and this is right! The gravitational force affects spacetime that the photons are traveling in so their trajectories will be bent!

AB

 

[xlines]

The acceleration of a mass towards another body is a byproduct of the curvature of spacetime.
If people argue this, then they don't understand it. Which will be common since four dimensions is one dimension bigger than our brain evolved to naturally conceptualise.



the concept of a force is a simplification we use of a physical process so our meagre intellects can interpret it e.g. Newtonian physics is a simplification of Einsteinian mechanics

For simplicity of calculations, we conform to the standardized F=mg

g=F/m

What you are saying it that you think gravity should be changed to units of kg.m/s/s and not the m/s/s that everyone else uses. In which case we would call it a force, and not gravity.

I am sorry, but I do have trouble understanding what you just said, since my native language is not English. However, I think you are implying that it's geometric nature is the reason why gravity is not force, but rather a background context. If this is the case, I am not convinced that gravity will retain it's geometric nature at physical regimes where quantum effects are relevant too; that is why I don't want to disqualify gravity as a force just yet. There are gravitational phenomena which can not be reduced to pure geometry - for an example, quantum effect of gravity-induced phase change which are theoretically predicted and experimentally demonstrated by neutron interferometry.

Would you, please, comment on how you see this phenomena.

 

[Nabeshin]

I'm sorry, but what happens to the spacetime to be flat? The gravitational field exerts a force on both particles and the fabric of spacetime and if it is not present, the spacetime reduces to a flat spacetime so all particles move along straight lines! If it is present, then the particles would have proper accelerations due to the force acting on them!

What causes spacetime to be flat is the absence of mass. The Einstein equations read:
$$G_{\mu \nu} = 8 \pi T_{\mu \nu}$$
Where $$T_{\mu \nu}$$ is the stress-energy-momentum tensor. When this is zero, the EE reduce to the vacuum Einstein equations, one solution to which is flat (minkowski) spacetime (Other solutions exist, which involve dirac delta functions. For example, a schwarzschild black hole). This is what I mean by spacetime being flat: the absense of any contribution from the stress-energy-momentum tensor. In this scenario, Newton's 2nd law is followed for free particles. The introduction of anything that makes the stress-energy-momentum tensor non-vanishing necessarily curves space-time, changing the christoffel symbols, and thus the EOM for the particle.

Please notice, I have not mentioned the word gravity even once. The word is not necessary in GR. Do not attempt to introduce it, and refer to the preface in my previous post.

 

[Altabeh]

What causes spacetime to be flat is the absence of mass. The Einstein equations read:
$$G_{\mu \nu} = 8 \pi T_{\mu \nu}$$
Where $$T_{\mu \nu}$$ is the stress-energy-momentum tensor. When this is zero, the EE reduce to the vacuum Einstein equations, one solution to which is flat (minkowski) spacetime (Other solutions exist, which involve dirac delta functions. For example, a schwarzschild black hole). This is what I mean by spacetime being flat: the absense of any contribution from the stress-energy-momentum tensor.

Please stop here! What comes to mind when hitting the stress-energy-momentum tensor in general relativity? You're saying that if $$T_{\mu\nu}$$ is zero, the spacetime is flat. Okay, but the story has not reached its end yet as you are missing two points:

1- In GR, every mass that curves spacetime has a gravitational nature but in general the tricky word "mass" can replace the widely- used 'stuff' from a physical point of view which has nothing to do with the curvature of spacetime unless we spacify whatever happens between these masses and the fabric of spacetime. You're just putting a cap on the name "gravity"!

2- The stress-energy-momentum tensor describes matter (density, pressure and stress), radiation and non-gravitational "force fields"! All these attributes belong to "gravitational field" (which exerts a gravitational force on the things around) in the Einstein's field equations and this field in turn may be inspired by the existence of "mass" as in the theory of Newtonian gravity.

Please notice, I have not mentioned the word gravity even once. The word is not necessary in GR. Do not attempt to introduce it, and refer to the preface in my previous post.

That is just because you're indirectly referring to it! I can also say something about a guy named "Baron Schilling Von Canstatt" but never quote his name clearly and rather use "he was a great guy who invented the telegram in 1832"!

AB

 

[Nabeshin]

I get the feeling we're arguing over semantics, or else I'm not understanding you clearly. My point is as follows: within the theory of GR (independent from Newtonian mechanics!) one never need mention a mysterious "gravity". All you need to mention is the stress-energy-momentum tensor.

I've re-read your post a dozen or so times and I really have a hard time understanding, I'm sorry. Could you, or perhaps someone else who understands, try and phrase it differently?

 

[TcheQ]

However, I think you are implying that it's geometric nature is the reason why gravity is not force, but rather a background context. If this is the case, I am not convinced that gravity will retain it's geometric nature at physical regimes where quantum effects are relevant too; that is why I don't want to disqualify gravity as a force just yet. There are gravitational phenomena which can not be reduced to pure geometry - for an example, quantum effect of gravity-induced phase change which are theoretically predicted and experimentally demonstrated by neutron interferometry.

Would you, please, comment on how you see this phenomena.

The phrase 'the force of gravity' is a misnomer. It is not a force. (gravity is acceleration, Newton defined "force" as being equivalent to acceleration multiplied by mass)

As others have stated, gravity does not exist in an Einsteinian world. If you are going to using gravity to explain things, and treat it as a force, then shows us some equations on how gravity affects zero-mass objects. (it doesn't work)

THe video I quoted shows at low speed, Newtonian mechanics is the same as Einstein Mechanics. The effect of gravity on an object with mass can be used to explain low-speed phenomena. It also shows at the beginning (if you watch the whole video) how the Newton equations are derived from the (1+$$\epsilon$$)ⁿ example

 

[Altabeh]

As others have stated, gravity does not exist in an Einsteinian world. If you are going to using gravity to explain things, and treat it as a force, then shows us some equations on how gravity affects zero-mass objects. (it doesn't work)

Again the gravitational field affects the fabric of spacetime through exerting a force on it and all particles such as photons moving in the curved spacetime will have their trajectories curved. This can also be seen for example from Rindler metric where spacetime is flat, but yet a uniform gravitational field exists locally to make the particles fall free by applying a uniform gravitational force on them.

AB

 

[Mentz114]

Newtonian physics can predict the deviation of light paths around massive bodies. The eclipse observations early in the last century determined that the value was closer to that predicted by GR than by the Newtonian method. The GR prediction is just double the Newtonian prediction.

See http://en.wikipedia.org/wiki/Tests_of_general_relativity for a discussion.

It was not until the late 1960s that it was definitively shown that the amount of deflection was the full value predicted by general relativity, and not half that number. The Einstein ring is an example of the deflection of light from distant galaxies by more nearby objects.

 

[Altabeh]

THe video I quoted shows at low speed, Newtonian mechanics is the same as Einstein Mechanics. The effect of gravity on an object with mass can be used to explain low-speed phenomena. It also shows at the beginning (if you watch the whole video) how the Newton equations are derived from the (1+$$\epsilon$$)ⁿ example

This is in agreement with some part of one of my posts here:

All these attributes belong to "gravitational field" (which exerts a gravitational force on the things around) in the Einstein's field equations and this field in turn may be inspired by the existence of "mass" as in the theory of Newtonian gravity.

The point is that nothing can be accelerated by itself. There must be a force acting on particles which in GR it is considered to be gravitational field! I don't know why that is such a big challenge for people to simply see why gravity is a force though I'm aware of the differences physicists have in the literature. Some believe it's no longer a universal force but curvature and distorsion in GR such as

The grip of gravity: the quest to understand the laws of motion and gravitation By Prabhakar Gondhalekar, page 161.

Define Universe and Give Two Examples By Barton E. Dahneke, page 167.

The search for gravity waves By P. C. W. Davies, chapter 2, section 1.

But all say that this is just because in GR one can't make use of a universal (Newtonian) force and can only retrieve such "force of gravity" locally. I agree with this but this does not imply the word "gravity" isn't a force in GR at all! Let's say it's not a universal force that we are used to encountering it in the classical physics.

AB

 

[Mentz114]

It is possible to think of gravity in GR as a 'pseudo-force'. The equations of geodesic motion state that

$$\frac{d \dot{x}^\alpha}{d \tau} = - \Gamma^\alpha_{\beta \gamma} \dot{x}^{\beta} \dot{x}^{\gamma}$$

from which

$$\delta \dot{x}^\alpha = - \left( \Gamma^\alpha_{\beta \gamma} \dot{x}^{\beta} \dot{x}^{\gamma}\right) \delta \tau$$

For suitable starting conditions for the velocities and positions of a test particle one can use this as the basis of a numeric solution.

The point being that here the Christoffel symbols act like a velocity dependent potential.

 

[TcheQ]

This is in agreement with some part of one of my posts here:

The point is that nothing can be accelerated by itself. There must be a force acting on particles which in GR it is considered to be gravitational field! I don't know why that is such a big challenge for people to simply see why gravity is a force though I'm aware of the differences physicists have in the literature. Some believe it's no longer a universal force but curvature and distorsion in GR such as

The grip of gravity: the quest to understand the laws of motion and gravitation By Prabhakar Gondhalekar, page 161.

Define Universe and Give Two Examples By Barton E. Dahneke, page 167.

The search for gravity waves By P. C. W. Davies, chapter 2, section 1.

But all say that this is just because in GR one can't make use of a universal (Newtonian) force and can only retrieve such "force of gravity" locally. I agree with this but this does not imply the word "gravity" isn't a force in GR at all! Let's say it's not a universal force that we are used to encountering it in the classical physics.

AB

An argument I posted for xlines is that dimensions need to be conserved.

i.e. Force=kg.m.s$$^{-2}$$ (joules/distance)
gravity =m.s$$^{-2}$$ (joules/distance/mass)

i.e. it doesn't fit the what we currently define as force, just by it's unit nature.

I think one of the consequences of using this approach means black holes evaporate, but that doesn't mean you can't use this wrong concept to solve day to day problems (it got used for 250 years just fine :).

 

[Gear300]

Do objects still accelerate due to gravity in GR?

 

[TcheQ]

How do objects accelerate in GR?

Fixed.

See Nebeshin's answer on page 1

 

[DrGreg]

For the benefit of readers who can't follow the technical jargon of this thread, in relativity acceleration and force are relative like lots of other things. Two observers can disagree over whether something is accelerating or not. As another contributor to this forum once said, to decide if something is accelerating, you first have to decide what is not accelerating, and measure things relative to that.

Relative to someone stood on planet Earth, falling objects accelerate and there is a force causing the acceleration. Relative to someone who is falling under gravity, another local falling object moves with constant (or zero) velocity and there is no force acting on it. This second point of view is particularly relevant: acceleration relative to a local, free-falling observer (momentarily traveling at the same speed) is called "proper acceleration" and it's what an accelerometer measures. Someone falling under gravity experiences zero proper acceleration. Someone stood on the Earth's surface experiences 1 g proper acceleration upwards.

I was careful to use the word "local" in certain places above, because "tidal forces" (a.k.a. spacetime curvature) can cause two free-falling particles which are a distance apart to accelerate from each other even though both have zero proper acceleration.

In GR, when people talk of acceleration they often mean proper acceleration; any other sort of acceleration is called "coordinate acceleration".

 

[Gear300]

It seems to me that objects still accelerate under the space-time curvature in GR...and I suppose we could also associate an energy with respect to this. So then what is the proper definition of force?

 

[StandardsGuy]

OK, let's cut the crap. Will somebody please explain why two objects with mass (which are not moving initially) start to move toward each other? It seems that would require force, whether or not you call it gravity.

 

[MikeLizzi]

OK, let's cut the crap. Will somebody please explain why two objects with mass (which are not moving initially) start to move toward each other? It seems that would require force, whether or not you call it gravity.

The following URL helped me visualize the curvature of spacetime.
http://www.adamtoons.de/physics/relativity.swf

 

[andrebourbaki]

Gravity is not a force. What is a force, anyway? Newton clarified for almost the first time in Science what a force is: First I will say it, then explain it: A force is something which makes the motion of a body deviate from uniform straightline motion.

Newton pointed out that bodies have a tendency, inertia, to continue in whatever direction they are already going, with whatever velocity they have at the moment. That means uniform, rectilineal motion: steady velocity, same direction. Newton actually knew this was what would be later called a geodesic, since « a straight line is the shortest distance between two points ».

It was then Einstein (and partly Mach before him) who said this does not get to the essence of the question. For Einstein, any coordinate system had to be equally allowable, and in fact, space-time is curved (as already explained by other posters). A body or particle under the influence of gravity actually does travel in a geodesic...i.e., it does what a free particle does. I.e., it does what a particle *not under the influence of any force* does. So gravity is not a force.

Newton did not realize that space-time could be curved and that then the geodesics would not appear to our sight to be straight lines when projected into space alone. That ellipse you see in pictures of planetary orbits? It is not really there of course since the planet only reaches different points of the ellipse at different times...that ellipse is not what the planet really traverses in space-time, it is the projection of the path of the planet onto a slice of space, it is really only the shadow of the true path of the planet, and seems much more curved than the true path really is. (The curvature of space-time in the neighbourhood of the Earth is really very small! The path of the Earth in space-time would even appear to be nearly straight to an imaginary Euclidean observer who, in a flat five-dimensional space larger than ours, was looking down on us in our slightly curved four dimensional space-time embedded in their world.)

Since every particle under the influence of gravity alone moves in a geodesic, it does not experience any force that would make it depart from its inertia and make it depart from this geodesic. So gravity is not a force, but electric forces still do exist. They could overcome the inertia of a charged body and make it deviate from the geodesic it is headed on: change its speed and direction (when speed and direction are measured in curved space-time).

Einstein (and me too) did not want to change the definition of force in this new situation, since after all electric forces are known to exist and are still forces in GR. So the old notion of force still retains its usefulness for things *other than gravity*. To repeat: if a body is not moving in a geodesic in space-time, you go looking for a force that is overcoming its inertia...but since gravity and curvature of space-time do not make a body depart from a geodesic, neither of them is a force.

 

[andrebourbaki]

It is possible to think of gravity in GR as a 'pseudo-force'. The equations of geodesic motion state that


$$\delta \dot{x}^\alpha = - \left( \Gamma^\alpha_{\beta \gamma} \dot{x}^{\beta} \dot{x}^{\gamma}\right) \delta \tau$$


The point being that here the Christoffel symbols act like a velocity dependent potential.

Quite right. But for some reason I was looking at Einstein's original papers the other day. Did you know he sometimes calls the Christoffel symbols « components of the gravitational field », sometimes the metric is called « gravity », and sometimes the curvature is called « the expression of gravity »? He was attacked in print for betraying his own principle of equivalence by introducing real forces of gravity, due to his calling the Christoffel symbols « gravity ». Can you imagine how he defended himself? By saying that well, callling them that was in principle « meaningless » and he only did it for the sake of continuity with our physical habits of thinking (from older theories). That is, he picked out the Christoffel symbols although they're not even a tensor, because they did describe the departure of the geodesic from an apparent straight line in space with respect to that coordinate system. And that is the definition of a pseudo-force, even in Newtonian Mechanics, whenever you use a classically disallowed non-inertial frame of reference. This puts a whole new perspective on the word « pseudo ».