# Understanding the General Relativity View of Gravity on Earth

**Important Concepts**

- Spacetime: the combination of 3 dimensions of space and 1 dimension of time into a unified 4D “manifold”.
- Coordinate systems: mappings between events in spacetime and 4 numbers called coordinates. Usually the coordinates involve 1 time coordinate and 3 spatial coordinates.
- Proper acceleration: the acceleration measured by an ideal accelerometer. This is the acceleration that you physically “feel”.
- Coordinate acceleration: the 2nd derivative of position in some given coordinate system.
- Inertial frame: a coordinate system where inertial objects have no coordinate acceleration. In inertial frames the line formed by an inertial object’s coordinates is a straight line.
- Curvature: when a surface is not flat, for example, the surface of a sphere. The geometry is different in curved manifolds versus flat ones. For instance, the angles in a triangle may not sum to ##\pi##.

**Non-inertial Frames and Fictitious Forces**

Physics is traditionally taught using inertial frames, however there is no restriction that prevents the use of non-inertial frames (e.g. a rotating reference frame). In a non-inertial frame inertial objects undergo coordinate acceleration, and in order to use Newton’s 2nd law in such frames additional “fictitious forces” (e.g. the centrifugal and Coriolis forces) are introduced.

These fictitious forces all share a few properties:

- They are proportional to the mass of the object
- They cannot be detected by an accelerometer
- They disappear in inertial frames

**The Force of Gravity: Newton vs. Einstein**

In Newtonian mechanics gravity is considered to be a real force, despite the fact that it shares the first two properties of fictitious forces listed. This makes Newtonian gravity a bit of a strange force. You cannot determine if a given reference frame is inertial or not simply by using accelerometers, you have to additionally know the distribution of mass nearby in order to correct your accelerometer readings for the presence of gravity.

In GR, this is simplified by considering gravity to be a fictitious force just like any other force which is proportional to the mass and cannot be measured by an accelerometer. This means that in GR an apple in free fall to the ground is considered to be inertial, while in Newtonian gravity the apple is non-inertial. Similarly, the rest frame of the free-falling apple is an inertial frame in GR with no fictitious forces, while in Newtonian mechanics it is a non-inertial frame with a fictitious force that is equal and opposite to the force of gravity.

**The Apple Falling**

As mentioned above, an apple freely falling to the ground is considered to be inertial in GR. If we make a coordinate system where the apple is at rest, it is an inertial coordinate system. In this reference frame, the apple does not accelerate, instead, the ground accelerates upwards at g and slams into the apple. This corresponds to the fact that an accelerometer attached to the apple reads 0, while an accelerometer attached to the ground reads 1 g upwards.

This is a bit odd. What is causing the ground to accelerate?

If you draw a free-body diagram of a small section of the ground, you find that there is a rather large real upwards pressure force on the bottom of the section, and this upwards pressure is not balanced by a corresponding large downwards pressure on the top. Therefore, there is a net upwards force on the ground, which is responsible for the upwards acceleration. Although this seems like a strange way to think, at first, it is clear that this approach still accounts for whether or not the apple splatters when it hits the ground, and so forth.

**Spacetime Geometry**

Now consider a “spacetime diagram” of the apple, where time is plotted on one axis and the apple’s vertical position is plotted on the other axis. In such a diagram the apple can be represented by a line (called a “worldline”) which shows its position at each point in time. If we take the frame where the apple is at rest then the apple travels along a straight line which is parallel to the time axis. A second apple that started free falling a moment earlier or later would have some constant velocity relative to the first apple, so its worldline would also be a straight line, but not parallel to the time axis.

So objects at rest relative to each other have parallel worldlines while objects that are moving relative to each other have non-parallel worldlines. Similarly, objects at rest relative to a reference frame have worldlines which are parallel to the time axis.

Now, consider a point on the ground. This point is initially at rest relative to the apple (and therefore the reference frame), so its worldline starts out parallel to the time axis. However, by the time the ground collides with the apple it has clearly gained some relative velocity and is no longer parallel. In other words, the ground’s worldline is curved.

So inertial objects (accelerometer reads 0) have straight worldlines while non-inertial objects have worldlines which are curved in a direction and by an amount given by their proper acceleration.

**Curved Spacetime and Tidal Gravity**

While this is all well and good for a single apple falling in a uniform gravitational field, what happens when we consider tidal gravity (gravity that varies over space) such as two apples falling on opposite sides of the world?

Let’s suppose that there is a hole completely through the Earth and no atmosphere, so that we can neglect the ground and the air (let’s also neglect the rotation of the Earth). In this case, the two apples will start out on opposite sides of the Earth and eventually collide when they reach the middle. When they start out they will be at rest relative to each other, and when they collide they will have a considerable velocity relative to each other.

How can that work with our spacetime view? We already established that being at rest means that their worldlines are parallel, and having relative velocity means that their worldlines are not parallel. So the two apples’ worldlines start out parallel and then wind up non-parallel. However, we also established that having 0 proper acceleration (i.e. being inertial and having 0 accelerometer reading) means that the worldline is straight. So both apples’ worldlines are straight. How can we reconcile the straight worldlines with the fact that they go from being parallel to intersecting? The answer is that in the presence of tidal gravity spacetime is curved, not flat.

A straight line in a curved space is called a geodesic, for example, a great circle is a geodesic on the surface of a sphere. A longitude line is a great circle, and therefore a geodesic. Consider two nearby longitude lines at the equator they are parallel but at the pole they intersect, despite being everywhere straight. So curved spaces have the necessary geometric properties.

Using this idea of curved spacetime, we can describe how the two apples can have 0 proper acceleration everywhere (straight worldlines) and yet accelerate relative to each other (initially parallel, but later intersecting). However, this has some consequences. The most critical is that there is no longer any such thing as an inertial frame which covers both apples, inertial frames are now “local” meaning that you can only use them when tidal effects are negligible (spacetime curvature is negligible).

**The Surface of the Earth**

In the previous section we neglected the ground, but now let’s consider the ground. If the ground is accelerating upwards and the direction corresponding to “up” changes around the globe, then it seems that the surface of the Earth should be expanding, with the distance from one point to another continually increasing.

This reasoning is incorrect in curved spacetime. In a flat spacetime it would be correct that the surface of the Earth could not be accelerating (proper acceleration) outwards while retaining a constant radius, but spacetime is curved and so it can indeed accelerate (proper acceleration) outwards while retaining a constant radius.

To understand the importance of curvature, consider two latitude lines on a sphere. For simplicity consider the latitude lines 5° N and 5° S. As you follow those lines around the sphere, they maintain a constant distance from each other. However, the 5° N line is constantly turning to the north and the 5° S line is constantly turning to the south. So they are turning away from each other but maintaining constant distance. This is impossible on a flat surface, but possible in a curved surface.

In the geometry of the spacetime around the Earth (i.e. Schwarzschild spacetime) you can take any two points on the surface of the Earth and find that they are accelerating (covariant derivative = proper acceleration) in different directions, and yet, because the spacetime is curved, the distance between them does not change (e.g. as measured by radar).

**Summary and Key Points**

- In GR the force of gravity is considered to be a fictitious force, and inertial reference frames are free-fall frames where falling apples have straight worldlines and the ground continuously accelerates upwards at g.
- In order to model tidal gravity (where gravity varies from location to location), we must use curved spacetime.
- This allows for two apples to each have straight (geodesic) worldlines but still accelerate relative to each other.
- It also allows for the ground to continuously accelerate upwards without the Earth expanding.
- However, it means that you cannot make an inertial frame which covers the whole Earth, and so inertial frames are only local.

[I could not find the two comments already posted]good write up….unsure of background education experience you are aiming at….accelerometer: maybe an explanation??….eg, it measures proper acceleration relative to free fall…Proper acceleration: "the acceleration measured by an ideal accelerometer" [consider adding: an acceleration an observer feels]Coordinate acceleration: "the 2nd derivative of position in some given coordinate system [add: an acceleration not felt]Inertial frame: a coordinate system where inertial objects have no coordinate acceleration [I thought an inertial frame had no proper acceleration.] [yes, you say this later:"So inertial objects (accelerometer reads 0)…….How about equivalence principle…That helped me at first….Nice insight:"In Newtonian mechanics gravity is considered to be a real force, despite the fact that it shares the first two properties of fictitious forces listed. This makes Newtonian gravity a bit of a strange force. You cannot determine if a given reference frame is inertial or not simply by using accelerometers, you have to additionally know the distribution of mass nearby in order to correct your accelerometer readings for the presence of gravity.

Cool post! I really liked how you explained the geodesic and ground's upward acceleration parts. I just didn't get the "free-body diagram of a small section of the ground" part. Can you elaborate on this a bit?

sorry i still don't get it. I understand that the north latitude line is turning in a circular path, but it's not really turning towards the north pole just towards the axis of the north pole. and the same with the south latitude line, and arent the north and south pole on the same axis? so doesnt that mean both latitude lines are turning in the same direction. or am i thinking this because i'm visualizing this in three dimensions? thanks.

Great post, DaleSpam!

Yes, very nice article. I’d only make sure to say once that gravity is not due to mass (energy) only (as in the Newtonian theory of gravity) but to all forms of energy-momentum distributions. This explains why light, which is described by massless spin-1 fields is affected by gravity (bending of light at the sun as one of the most important early tests of GR; red shift of light in gravitational field) and (in principle) is a source of gravity itself.

I don’t understand what it means that ” 5° N line is constantly turning to the north and the 5° S line is constantly turning to the south”.

Could someone explain?

If it is hard to see at first then consider the 89.9 degree latitude line. This is a tight little circle around the pole, so to stay on the latitude line you have to constantly turn towards the pole.

The same thing happens on the 5 degree latitude line, it just is not as tight of a turn.

I get it now, thanks.

And now I can thank you a lot for the insight article, because this was the only thing that was keeping me from understanding this issue. So thanks.

That is a good idea, but I am not sure it is a good idea for an “everyday gravity” explanation. I also avoided any discussion of time dilation for the same reason.

I will look back and see if there is a good place to put that in without much distraction.

I like that. I will add that.

You do “feel” coordinate acceleration in a GR local inertial frame (since it is equal to proper acceleration).

I did think about wording similar to that, but the problem is that Newtonian and GR inertial frames are different. In GR inertial frames have no proper acceleration, but in Newtonian mechanics inertial frames have a proper acceleration of -g. I tried to word it in a way that is true for both.

It could probably still use some improvement, but a theory-neutral explanation is difficult.

Thanks. I appreciate the encouragement.

The good textbooks that I know clearly differentiate between “inertial motion” and “inertial frames” on the one hand, and “local inertial frames” on the other hand. Those mimic inertial frames for sufficiently local measurements. There is as a consequence a consistent use of terms throughout those textbooks, independent of theory.

To avoid unnecessary confusion it is better to follow that example: the rest frame of the free-falling apple is a “LOCAL inertial frame” in GR, so that the apple can be considered as “inertial” locally.

PS: Einstein had a subtly different view of GR than the view that you describe as “the GR view”, and surely he also taught GR. And Lorentz again had a subtly different view, and he also taught GR. In fact GR is interpretation neutral, as it is foremost mathematical, making predictions of observations. What you describe is perhaps more correctly indicated as the geometric view of GR, or the Minkowskian view of GR.

Lorentz had also a different view concerning SR. Fortunately this is overcome in the physics community, and there is a unique view about relativity. Unfortunately, one can’t say this about QT, where in some niches of the scientific universe there coexist very different interpretations and metaphysics (reaching well into the realm of esoterics), and I’m not talking about obvious crackpots ;-)).

The problem for this description is not that GR inertial frames are local and Newtonian inertial frames are global. The problem is that even locally they disagree. So stressing “local” doesn’t avoid the reason that I chose that description.

There are many equivalent ways of defining an inertial frame. I chose one that I thought fit best with the intention of the article.

Sure. Consider a 1 cubic meter chunk of soil. If we draw a free-body diagram of that chunk of soil then we have real pressure forces on all 6 faces of the cube. The left and right and the front and back pressures all cancel out. However, the pressure force on the top is much less than the pressure force on the bottom, so they do not cancel out and there is a net pressure force upwards.

In the Newtonian inertial frame, that upwards pressure force is exactly balanced by the downwards gravitational force.

In the GR inertial frame, the downwards gravitational force does not exist, so the upwards pressure force is unbalanced and causes the ground to accelerate upwards.

You can approximate a small latituderange with a cone:

[URL]https://en.wikipedia.org/wiki/Map_projection#Conic[/URL]

If you roll out the cone flat, you end up with this local picture:

Ah, got it, thanks!

Not really: even Newton’s mechanics recognized local inertial frames as follows:

[I]”If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another

in the same way as they would if they were not acted onby those forces.” (emphasis mine)[/I]

The pertinent difference for physics (that is, leaving aside philosophy and nomenclature) is that GR postulates this equivalence not only for Newton’s mechanics but for all physics.

Why would it be “fortunate” if there is a unique metaphysical opinion in the physics community? Physics must be based on facts of observation. Consequently the situation with QT is perhaps better – except from the esoterical part! o0)

According to your interpretation of this definition, which frame is inertial:

– A frame at rest to the surface of a non-rotating planet?

– A frame free falling towards that planet?

– Both?

You are reading something into this that simply isn’t there. Neither the word “local” nor “inertial” nor “frame” even appears.

To me this quote seems to be describing the use of non-inertial frames to eliminate real forces and simplify an analysis, although it isn’t using clear terminology so I cannot be certain. I see no mention of anything local.

What is the source for this quote? I am guessing that it is something quite old, before the terminology became clarified. I believe that my presentation accurately reflects the modern usage, and it is not intended to be an historical treatise.