# Question about inertial frames and geodesics

Consider the following text:

In Newtonian Mechanics the first law is given by: $$\Big[m \Big(\frac{d^{2}x^{a}}{dt^{2}} + \Gamma^{a}_{bc}\frac{dx^{b}}{dt}\frac{dx^{c}}{dt}\Big)\Big]\frac{\partial}{\partial x^{a}} = \vec{0} \tag{1}$$

The equation $$(1)$$ says that with respect a certain class of reference frames a body with a null net force remains at rest, or moves under (a notion of) straight lines (solutions of the non-linear coupled differential equation above).

Example:

In cartesian coordinates the levi-civita connection symbols are all:

$$\Gamma^{a}_{bc}= 0$$

then $$(1)$$ becomes,

$$\Big[ m\Big( \frac{d^{2}x^{a}}{dt^{2}}\Big)\Big]\frac{\partial}{\partial x^{a}} = \vec{0}$$

And the solutions are (the notion of straight lines):

$$b+at = f(t)$$

Now,the following statements are equivalent:

1) That class of reference frames are called inertial frames. In that frames we can distinguish bodies moving under a non-geodesic path from those who are.

2) The equation $$(1)$$ is the law of Inertia. A body $$S$$ that satisfies the equation $$(1)$$ defines a inertial reference frame. Then the motion of another body $$C$$, described with respect to $$S$$, can be at rest,uniform or accelerated.

3) A body that moves under a non-geodesic path, defines a non-inertial frame of reference.

My question is: Why the 1),2) and 3) are equivalent?

• Can you be more specific about what you find unclear about these 3 statements, say, by attempting to show the equivalence of two of them and showing where you get confused? – CR Drost Nov 17 '18 at 4:23
• @CRDrost "The equation (1) is the law of Inertia. A body S that satisfies the equation (1) defines a inertial reference frame. Then the motion of another body C, described with respect to S, can be at rest,uniform or accelerated." Is this phrase right? – M.N.Raia Nov 17 '18 at 4:32

1) That class of reference frames are called inertial frames. In that frames we can distinguish bodies moving under a non-geodesic path from those which are.

If a body is at rest or moves at constant velocity (i.e. $$\sum\vec{F} = 0$$) it is located at an inertial frame.

Indeed, we can distinguish bodies moving between non-geodesic and geodesic paths at an inertial frame. To do so I would use this test:

As we know, a geodesic is a curve along which the tangent vector is parallel-transported. Based on this statement, try to transport a tangent velocity vector in a parallel way. If you succeed means your vector takes a geodesic. Otherwise your vector does not. Such a case would confirm that your body is not located at an inertial frame. This is because your tangent velocity vector would change its direction (you have just checked that your vector cannot be parallel-transported), meaning that your body would not have constant velocity so that it is not located at an inertial frame.

2) The equation (1) is the law of Inertia. A body S that satisfies the equation (1) defines a inertial reference frame. Then the motion of another body C, described with respect to S, can be at rest,uniform or accelerated.

This text just describes more features related to an inertial frame.

3) A body that moves under a non-geodesic path, defines a non-inertial frame of reference.

I have explained this at 1)

Why are these statements equivalent?

These describe the motion of bodies which go through either geodesics or non-geodesics paths and subsequently if they are located at an inertial frame or not (respectively).