General Relativity - Why You Are Not Accelerating When You Fall!

General Relativity Part 2

This is the second part of this general relativity mini series. I recommend that you read part 1 before continuing with this post.

Why Do We Fall?

Remember that we are moving through space-time at the speed of light? Well we are also moving on the path of least resistance (geodesics) through space-time. In flat space-time where there is no gravity, this is a straight line. Gravity bends space-time and so when we come near large objects, we begin to move through curved space-time.
This causes us to move towards the large object as this is the path of least resistance and so this is why we fall. This path has no forces on it and this is why free-fall feels like inertial motion. The path is also straight and straight paths through space-time indicate inertial observers and so free-fall is actually inertial motion. (If the straight path part confuses you look at the definition of straight lines and what it means for curved spaces).

Imagine if you rolled a ball in a straight line on a curved surface. The ball would change direction and bend towards the curved object. This is analogous to what happens in 4D space-time. 4D is impossible to visualize but the ball analogy works by giving you a feel for what warped space looks like. The picture below is the kind you will see when looking at general relativity and it can help give an idea of whats going on.

This Doesn't Make Sense? Free-fall is definitely acceleration

Newton would claim that all inertial observers can agree that each other are inertial. If you have a group of people moving around deep space at a constant speed, they can all agree that none of them are accelerating; they will just measure each other moving at different constant velocities. He would say that it is obvious who is accelerating because these people are changing velocity.
An example of this is that if an apple then accelerates towards the Earth; all the observers can agree that the apple is accelerating. How can Einstein justify that the apple is not accelerating?

The equivalence principle

The outcome of any local experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in space-time.
General relativity puts all observers on an equal playing field. All observers who do not feel a force acting on them can claim to be inertial even if they are in a gravitational field. This means the outcome of an experiment while you are in free fall is the same as if you are stationary.
Local is the key word here however. The experiment must be local in space-time (in a small space over a small period of time).

In the experiment above, the balls fall towards each other. If you were in the freely falling laboratory however, you would see the balls stay at the same height but move towards each other. This is definitely not the result you would obtain if you did the experiment while stationary in deep space. In deep space, the balls would hover in the same place and not move relative to each other.

Local!

The different results is because this is not a local experiment. The gravitational field of Earth is not homogeneous and so it varies across space. This means the balls appear to move towards each other since the gravitational field acts differently on the different balls. As the lab becomes smaller and smaller, the variations in the gravitational field (tidal forces) decrease and the field becomes more like a homogeneous one. This means in an experiment local in space, the result will be the same as if the experiment was done while moving at a constant velocity. The key is reducing the space so that the tidal forces are reduced.
Locality in time is also important here. If the experiment proceeds for too long, you will start to notice that the balls fall radially. If you do it over very short periods of time however, you do not notice the balls falling radially and the outcome of the experiment is identical to one performed by an inertial observer.

The equivalence principle then allows us to have inertial reference frames everywhere in the universe (provided they are local in space-time). This means special relativity can be extended to include gravitational fields and we have a framework that works for the entire universe.

Back To Free Fall

So how would Einstein dispute the fact that free fall is acceleration?
He would say that you have to remember the equivalence principle.
Imagine a sphere. If you try to put a x,y grid flat on a sphere it would not work because the surface is curved. Try this by putting a piece of paper on a ball. The paper becomes all bunched up and if you attempted to use the paper as a coordinate grid, your coordinates wouldn't make much sense.

If you try make the paper smaller and then do the same, it starts to work better and better. If you keep your grid local, on a very small patch and not try to measure across patches, you can use your flat gird perfectly well. Try to use a massive piece of paper and everything goes wrong. Well this is what Einstein claimed Newton was doing. Einstein claimed that space-time was curved and so not keeping measurements local meant his statements were invalid. The same way it seems absurd to use a flat piece of paper to measure locations on opposite sides of a sphere is the same way it is absurd to for observers in deep space to claim apples are accelerating due to gravity. The observer is pushing his measurements past the point of reliability.

A consequence of this is that global inertial reference frames do not exist. Space-time is curved and so you can't have one flat sheet of paper as a coordinate system for the entire thing. You can have local inertial reference frames however and these must not push themselves past the point of reliability.

Why is the Earth not expanding radially?

Earlier we established that gravity and acceleration are indistinguishable and now we know that free fall is inertial motion. So this means that what we thought was objects accelerating downwards, was actually us being accelerated upwards. So the Earth is expanding radially?

Remember the equivalence principle? To compare two sides of the Earth, you need a inertial reference frame that stretches across the Earth, but these do not exist due to the curvature of space-time. So any conclusions we draw are using non inertial frames of reference. We know that the laws of physics are the same in all inertial frames of reference but what about non inertial? Well we never mentioned those because conclusions in accelerating frames must be taken with a heavy grain of salt.

Compare this to most of the experiment we carry out in labs. These do not take a very long time and do not span over large areas and so tidal forces are not very large. This is why we can perform experiments properly on Earth with few issues. The experiments we do are local (just like using a small piece of paper as a coordinate system on a ball).

Einstein's theory shows that the laws of physics are the same for all observers, even those undergoing accelerated motion (as long as they keep to their small patch of space-time). This can be done because accelerated observers are justified to claim that they are at rest and that the fictitious force he/she feels is due to a gravitational field.

So there it is. Einstein's theory of general relativity actually comes as a direct consequence of special relativity. It may seem pretty odd at first, but as you start to get a better understanding of relativity, you may come to the point where you think these things are actually obvious.
You may be able to reach a stage where you can start at special relativity's two postulates and logically follow through to the conclusion that gravity is the curvature of space-time.

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Did you see my previous post? Click the link below to check it out

GENERAL RELATIVITY - FREE-FALL, FICTITIOUS FORCES AND THE WARPING OF SPACE-TIME