# If gravity is additive, then how does it cancel itself out?

I understand that gravity, as far as we know, is always attractive. Also, it has additive qualities - i.e. the size and strength of the field are proportional to the quantities of mass.

This seems to counteract the idea that gravity can cancel itself out. The centre of the Earth is said to be a zero-g environment, yet it is in the midst of a whole load of mass. I guess this makes sense when thinking about the mass as pulling equally from all directions... Which leads on to two questions.

1. If opposing masses can effectively cancel each other out, does this mean Gravity is not always additive?

2. Is spacetime geometrically indistinguishable in an area of zero-g, lets say, between galaxies, and in the centre of very massive bodies, like a planet? What I mean here is, can you tell that there are strong gravitational forces pulling you in all directions as opposed to weak ones?

• Its additive in the sense that F(two masses)=F(one mass alone) + F(the other mass alone). Say im in the middle and one mass to each side of me. both would attract me but the net force is zero, this does not conflict with additivity – Bort Feb 10 '16 at 11:00
• Opposing forces can cancel out. Does this mean forces are not additive? Of course not - forces add as vectors, by the parallogram rule. – Peter Diehr Feb 10 '16 at 11:02
• Gravity is not even a force, even though we formulate it that way. The reason why this matters is because while forces are being characterized by a charge (electric charge, magnetic moments etc.), mass does not play this role for gravity. That is why there are no negative masses. While in the Newtonian limit gravity looks very similar to a force, in general relativity it behaves very differently and becomes both non-linear and non-conservative (because of gravitational waves), which leads to consequences that take the form of geometric and thermodynamic properties. – CuriousOne Feb 10 '16 at 11:19
• Second part of question is much interesting than first. You should have posted question separately. – Anubhav Goel Feb 10 '16 at 14:33

You need to keep the direction in mind. While the direction never makes the gravity negative, adding opposite directions will cancel out.

I don't know if you are familiar with vectors? The length of a vector is always positive (strength of gravity) but it also has a direction (direction of gravity). If you add two vectors of equal length ("strength") but with opposite direction they cancel out.

1. Gravity is always additive. You can never add mass and reduce the gravitational force created. What does happen is that gravitational forces in opposite directions can cancel one another out, leaving 0 net force. If gravity wasn't always additive, it would mean that it would be possible for gravity to push things, instead of always pulling. This is not the case. It seems the confusion here is stemming from vector addition - adding a 1N force pulling left to a 1N force pulling right gets you a 0N net force. The forces are still added together, it just so happens that the opposite directionality reduces the magnitude of the final net force.

2. Not sure what you mean by this question. The center of a planet is a 0 G environment, but spacetime is still "geometrically distinguishable". Moving away from the center in any direction causes you to feel a pull back toward the center. It's not like directionality or distance loses meaning in a 0 G environment.

• 1. but, semantically, if something can cancel itself out, that means it is not always additive. Always attractive, yes. 2. I edited the question above to make it more clear. – Amphibio Feb 10 '16 at 11:00
• Additive allows for negative numbers 1+(-1)=0. That is still additive. If there are two identical planets and you are standing in the middle between them the force on you is zero BECAUSE the gravitational effects are additive. It's simply because the pull from one is minus the pull from the other. – Ymareth Feb 10 '16 at 11:17
• @ymareth, does this mean gravity has polarity? i.e. 1 and -1? I think I'm going to ask a new question related to this – Amphibio Feb 10 '16 at 11:21
• Directionality, not polarity. You can define "to the right" as the positive direction, and "to the left" as the negative direction. If you have 1N pulling you right, it's a force of 1. A 1N force pulling you left is a force of -1. Adding them up gets you 1 + (-1) = 0. Look into vector addition - forces aren't just numbers, they're numbers with an associated direction. – Nuclear Wang Feb 10 '16 at 11:24
• Put it another way: if we assume gravity has polarity (which it doesn't as far as every experiment ever performed goes), in your example which of the masses is "pushing"? Neither, they're both "pulling". (Quotes used as "pull" and "push" aren't completely accurate terms to use.) That's the whole point. They're both pulling in opposite directions, but they're PULLING. You're suggesting gravity pushes in some circumstances. – The Geoff Feb 10 '16 at 13:41

I suspect you're getting confused about the difference between the gravitational potential energy and the gravitational force.

Potential energy is always additive, however in electromagnetism it can have different signs while in gravity it cannot. With electromagnetism the potential energy can be positive or negative so it is possible for the potential energy from two sources to cancel out and be zero.

By contrast, in gravity the potential energy is always negative so combining any two sources can only ever decrease the gravitational potential energy. The only way for gravitational potential energy to cancel would be if negative matter existed.

In both cases the force is the gradient of the potential energy:

$$\mathbf{F} = \nabla V$$

and the force can in principle have any magnitude and point in any direction.

Take your example to an extreme to see what happens - sit yourself just between two black holes. There will always be a (Lagrange) point where you're not pulled towards one or the other, but there will be pulls from either side...at an extreme, you'll be ripped in two.

"Cancelling out gravity" is about gradients - remember, the gradient on a graph can be zero even for vastly different Y-values.

As an analogy, you're asking: "I was at sea level and the ground was flat. Then I went up a mountain and it was flat at the top. How can the top of a mountain be at sea level?"

• I think this is at least misleading. The fact that you will be ripped apart is due to the fact that you are an extended object. The forces do NOT cancel all over your body, only in the Lagrange point. For a point mass sitting at the Lagrange point, the forces cancel perfectly and there is no way of telling that 2 BHs pull on you. No forces means you don't feel anything, doesn't matter if 2 forces cancel or there aren't any at the first place – Noldig Feb 10 '16 at 13:21
• Yup, fair comment. Although if we take the argument to the extreme, there are no point masses, and we hit the problem of describing quantum gravity, which is a little non-trivial ;) – The Geoff Feb 10 '16 at 13:24

There are two elements you need to keep in mind: direction and intensity. Direction is obvious, the intensity is the length of the vector. The fact that Gravity is always addictive means that aligned forces are addictive, therefore the lengths, if aligned and pointing in the same direction, will add up. There is no repulsive force of gravity, Gravity is always attractive. However you need to learn how to add up vectors that are not aligned. There is a rule to add vectors (and I emphasise, add, since gravity is always addictive) and if two vectors are aligned, same intensity, and pointing in opposite direction, their sum will be zero. In general, given two vectors, their sum is given by a third vector created by creating a parallelogram with equal opposite edges corresponding to the two vectors. The sum will then be given by the diagonal drawn in this parallelogram.

(1) Gravity is not additive it is constant but mass and acceleration due to gravity are additive. (2) I don't think you can tell the difference by force but maybe from time dilations.