# Coordinate acceleration without a Force

In summary, there are two types of motion presented by GR: geodesic and non-geodesic. The equations for these motions involve the four-momentum derivative and the calculation of coordinate acceleration. The equation for coordinate acceleration can be simplified if there is no force present. However, there has been no response to this topic in over a month.

Coordinate acceleration without a Force !

Hi
GR had presented two types of motion , the geodesic motion and the non-geodesic motion . We know that the geodesic motion equation is :

$$$\frac{{d^2 x^\alpha }}{{d\tau ^2 }} + \Gamma _{\beta \mu }^\alpha \frac{{dx^\beta }}{{d\tau }}\frac{{dx^\mu }}{{d\tau }} = 0$$$
and ( as I know ) the equation of the non-geodesic motion in the derivative of the four - momentum :
$$$F^\lambda = \frac{{dP^\lambda }}{{d\tau }} + \Gamma _{\mu \nu }^\lambda U^\mu P^\nu$$$
and from this equation we can calculate the equation of the coordinate acceleration :
$$$a^\lambda = \frac{1}{m}\left( {\frac{{d\tau }}{{dt}}} \right)^2 \left[ {F^\lambda - \left( {\frac{{u^\lambda }}{c}} \right)F^0 } \right] + \left( {\frac{{u^\lambda }}{c}} \right)\Gamma _{\mu \nu }^0 u^\mu u^\nu - \Gamma _{\mu \nu }^\lambda u^\mu u^\nu {\rm }...{\rm Eq(1) }$$$
but I saw and worked with a different equation for the coordinate acceleation :
$$$a^\lambda = \left( {\frac{{u^\lambda }}{c}} \right)\Gamma _{\mu \nu }^0 u^\mu u^\nu - \Gamma _{\mu \nu }^\lambda u^\mu u^\nu$$$
and I saw that this equation is Eq1 with zero four - force . without force the motion won't be non-geodesic any more .
How ?
thanks

Hello !
It's been nearly a month since I put the theard and I didn't find any replies .
Is there is anything wrong in it ?

Thank you for your question. The coordinate acceleration without a force can be seen in the geodesic motion equation, where the acceleration is described by the Christoffel symbols (Γ) and the velocity (dx/dτ) of the particle. This equation describes the motion of a particle in a curved spacetime without any external forces acting on it. In this case, the acceleration is due to the curvature of spacetime itself, rather than a force.

In the non-geodesic motion equation, the coordinate acceleration is also described by the Christoffel symbols and the four-momentum (P). However, in this case, there is also a term for the force acting on the particle. When there is no force, this term becomes zero and the equation reduces to the geodesic motion equation.

In other words, the coordinate acceleration without a force is simply the geodesic motion equation, where the acceleration is purely due to the curvature of spacetime. This is a fundamental concept in general relativity, where the motion of particles is described by the geometry of spacetime rather than external forces. I hope this helps clarify the concept for you.

## 1. What is coordinate acceleration without a force?

Coordinate acceleration without a force is the acceleration of an object in a particular direction without the presence of an external force acting on it. This type of acceleration occurs when there is a change in the velocity of an object due to a change in its direction or due to the presence of an internal force, such as friction.

## 2. Why does coordinate acceleration occur without a force?

Coordinate acceleration occurs without a force because of the principle of inertia, which states that an object at rest will remain at rest and an object in motion will continue to move at a constant velocity unless acted upon by an external force. In the absence of any external forces, an object will continue to move in a straight line with a constant velocity.

## 3. How is coordinate acceleration without a force different from normal acceleration?

Coordinate acceleration without a force is different from normal acceleration because it does not require an external force to cause a change in the object's motion. Normal acceleration, on the other hand, is caused by external forces and results in a change in the object's speed or direction of motion.

## 4. Can an object experience coordinate acceleration without a force?

Yes, an object can experience coordinate acceleration without a force. This can happen when an object is moving in a curved path, as its velocity is constantly changing in direction, causing it to experience a change in acceleration without the presence of an external force.

## 5. How is coordinate acceleration without a force related to Newton's first law of motion?

Coordinate acceleration without a force is related to Newton's first law of motion, also known as the law of inertia. This law states that an object will remain at rest or continue moving with a constant velocity in a straight line unless acted upon by an external force. This means that when there is no external force acting on an object, it will experience coordinate acceleration without a force, as it follows the principle of inertia.