# What will I feel if I experience a force in an accelerating frame of reference?

If I am in an inertial frame of reference, and I experience a force, what will happen is that I will accelerate relative to this inertial frame of reference. Say I am 60 kg and the force is 5 N and initially, I am at rest relative to this inertial frame of reference. This can be described by Newton's laws of motion. My acceleration is 1/12 m/s^2, relative to this inertial frame of reference.

However, if I am in an accelerating frame of reference with a constant acceleration, with absolutely nothing around me, I wouldn't be able to tell I am accelerating. (Imagine I am falling in a uniform gravitational field and the frame of reference is also accelerating with me. A good example will be a falling bottle of water and I am viewing from the angle of one water molecule). In this case, if I am 60 kg and experience a force of 5 N and initially, I am at rest relative to this accelerating frame of reference (imagine that, for some reason, I experience some sort of mysterious stuff that makes me at rest relative to this accelerating frame of reference), what will happen? What is my acceleration relative to this accelerating frame of reference? Do the physics laws still apply and why? Do Newton's second law and third law apply and why?

Then, my next question would be: What will happen if the conditions are the same as stated above, apart from that the frame of reference is accelerating with a changing acceleration. And why?

## 3 Answers

Gravity is a special case because it acts on all local masses so as to effect the same acceleration, exactly like the pseudoforces associated with being in an accelerated frame. One can deal with this in one of two ways:

1. define space and time such that a little 4-dimensional brick of spacetime 1 light second by 1 light second by 1 light second by 1 second over here can be picked up, moved sonewhere else, and slotted into a grid of such squares anywhere else, in which case Gravity is a real force that just happens to work out like a pseudoforce, and accelerometers read the sum of all real forces except gravity, or

2. define space and time such that an accelerometer reads the sum of all real forces, but that brick of spacetime that you measured out over here won't always fit neatly in the grid over there, in which case falling frames are inertial and gravity is a pseudoforce.

It turns out that the second description, once mathematically formalized as a theory, is more generally useful for making predictions and ends up being mathematically simpler in the long run than treating gravity as a real force and adding various corrective terms to capture the behavior of clocks. So we've used that since the mathematics were figured out about a hundred years ago, and gravity is treated as just another pseudoforce.

In either case, your accelerometers - including the sensed biological ones such as the fluid displacement in your inner ear - will display or convey to your senses the vector sum of all real non-gravitational forces, including (especially) the force keeping you from falling through the ground to the center of the earth right now.

The sensory version of an accelerometer display is the sensation of weight, the orientation of "down", and the experience of being not falling.

If your rest frame is the same as the accelerating frame, it will be as if there is a body force acting on all objects in your frame, with an acceleration (like g) in the direction opposite to the frame acceleration with respect to an inertia frame. So, in your first example, if you are moving with the accelerated frame of reference, it is as if the 5 N force exerted on you is balanced by a body force of 5 N in the opposite direction, and you will imagine that you are at static equilibrium.

A more appropriate example will be; consider Bob and Alice in two separate identical cars. Both cars are initially at rest. Both of them accelerate at the same time. Now Bob thinks Alice is at rest as pointed by you in your example. But won't Bob feel a push backwards and will stick to his seat? This means Bob can clearly say he is being accelerated. Same follows for Alice too.

At this point, you may ask why didn't your example give the same result? Why didn't you feel any push while you were falling?

The answer, as pointed by others, comes down to the fact that gravity is a special case. And yes, the free falling frame is inertial(locally). Read this similar question.