How Gravitational force differs in different dimensions.
F=G m1m2/r^2, is this for all dimensions or only for three dimensional space?
for 4 dimensions would it be F=G m1m12/r^3
for 5 dimension - F=G m1m12/r^4
and so on
is it correct?
so what it would be for two dimensional and one dimensional.
would it be same as three dimensional? or value of r will be change for 1 and 2, I mean r for two dimensional and r^(0.5) for one dimensional.
F=G m1m2/r^2, is this for all dimensions or only for three dimensional space?
for 4 dimensions would it be F=G m1m12/r^3
for 5 dimension - F=G m1m12/r^4
and so on
is it correct?
so what it would be for two dimensional and one dimensional.
would it be same as three dimensional? or value of r will be change for 1 and 2, I mean r for two dimensional and r^(0.5) for one dimensional.
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Gravitational force, just like EM, in 2D falls linearly; in 1D it's constant.
Gravity in any number of dimensions can be derived using Gauss's theorem, which relates the flux of gravitational field lines through a multidimensional sphere with the mass inside the sphere. That gives gravitational potential,
which is GM/r in 3D, GM log(r) in 2D and GMr in 1D. The force is given by the derivative of the potential times the mass of the second body, i.e. F = GMm/r^2 in 3D, GMm/r in 2D and GMm in 1D.
Gravity in any number of dimensions can be derived using Gauss's theorem, which relates the flux of gravitational field lines through a multidimensional sphere with the mass inside the sphere. That gives gravitational potential,
which is GM/r in 3D, GM log(r) in 2D and GMr in 1D. The force is given by the derivative of the potential times the mass of the second body, i.e. F = GMm/r^2 in 3D, GMm/r in 2D and GMm in 1D.
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General relativity rigorously proves that the orbit an object takes when under free fall conditions in a gravitational field, is a geodesic in curved space-time.
In other words, if you bend a strait line, the effect of bending the line is that the 1 dimensional line is bent into 2 dimensions. Same thing with a curved 2d surface being bent into 3 dimensions. If you say space-time is 4 dimensional, then it can be proven mathematically that curved space-time occupies a minimum of 5 dimensions. Gravity is not a dimension of space itself, but a result of it being curved upward into the next dimension.
In other words, if you bend a strait line, the effect of bending the line is that the 1 dimensional line is bent into 2 dimensions. Same thing with a curved 2d surface being bent into 3 dimensions. If you say space-time is 4 dimensional, then it can be proven mathematically that curved space-time occupies a minimum of 5 dimensions. Gravity is not a dimension of space itself, but a result of it being curved upward into the next dimension.
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Newtonionly speaking, yes. (except r^0 for one dimension)
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Oh and don't forget libration points.
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i think it will be gm1m2/r^2 for all dimensions