Question regarding a Newtonian equation modification

[TRB8985]

Good afternoon all,

I'm going to sound like a blithering idiot in attempting to describe my question, so please forgive me. I appreciate your patience.

While working on problems in the gravitation chapter in my college physics textbook, I came across a very interesting situation that I don't have the expertise to reconcile, nor can easily find a solution to on Google. It isn't with a particular problem, but rather with the concept of blending a Newtonian and relativistic concept.

Take, for example, the familiar equation for computing the gravitational force between two objects: Gmm'/r^2.

My question is this:

Since distances between two objects are 3D vectors in "real" space, wouldn't it be more accurate to replace a one-dimensional length (like r) with something more akin to sqrt(x^2 + y^2 + z^2 + ct^2)? Thereby making the equation:

Gmm'/(sqrt(x^2 + y^2 + z^2 + ct^2))^2

I get the feeling that these two concepts may be extremely distant from one another in usability like that, but I'm unfortunately unable to reach any of my professors over the summer and can't speak to someone who would actually know better.

Thank you for your time. Enjoy the holiday weekend.

 

[axmls]

First: why the ct term? In relativity, this term is negative, but for Newtonian physics, I'm not sure why you put that in. Regardless, using r instead of the distance formula is more generalized. It's independent of the coordinate system that you choose. Sure, it works if you're in 3-D Cartesian coordinates where you have x, y, and z coordinates, but what if you're in spherical coordinates, where you represent a point using a radius and two angles? Or cylindrical coordinates, where you represent a point using a radius, an angle, and a height?

The point is: using r is better, because it's more general. In order to use the distance formula, we would need to know the Cartesian coordinates of the point. If we're working in spherical or cylindrical coordinates, then we would have to convert the points to Cartesian to use the formula. We don't have to do that if we use r, as r is just a generalized distance between the two points.

 

[Drakkith]

Technically it would be the square root of ((x2-x1)^2 + (y2-y1)^2...

This gives you the distance r, and the formula is so basic that you don't need to include it in the gravitational force equation.

 

[mic*]

https://en.m.wikipedia.org/wiki/General_relativity

The section "From classical mechanics to general relativty" may be useful.

 

[Josh S Thompson]

I agree

Since distances between two objects are 3D vectors in "real" space, wouldn't it be more accurate to replace a one-dimensional length (like r) with something more akin to sqrt(x^2 + y^2 + z^2 + ct^2)? Thereby making the equation:

It not one dimensional because its r^2