# B Why do bodies in free fall change position

#### Charly

Why do bodies in free fall change position relative to each other?

For instance in a vacuum with no external forces why can an apple fall towards the surface of the earth? If these objects aren't accelerating then are they considered at rest? And in what sense of the term 'rest'.

I think I can intuit how in a curved space time, WHEN a body experiences a force its movements are dictated by the curvature of the spacial geometry it moves through. I think my problem is I am only intuiting a curved space, rather than a curved space time.... Is it even possible to intuitively understand GR?

(To clarify: I'm just asking for help with the primary question, the rest was to attempt to give context to my thought to help reveal the issue in my understanding)

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#### berkeman

Mentor
Why do bodies in free fall change position relative to each other?

For instance in a vacuum with no external forces why can an apple fall towards the surface of the earth? If these objects aren't accelerating then are they considered at rest? And in what sense of the term 'rest'.

I think I can intuit how in a curved space time, WHEN a body experiences a force its movements are dictated by the curvature of the spacial geometry it moves through. I think my problem is I am only intuiting a curved space, rather than a curved space time.... Is it even possible to intuitively understand GR?

(To clarify: I'm just asking for help with the primary question, the rest was to attempt to give context to my thought to help reveal the issue in my understanding)
Welcome to the PF.

You don't need curved spacetime for this. The classical Law of Gravitation works just fine here. Are you familiar with that equation? If not, try Wikipedia and let us know if you still have questions.

#### Charly

Welcome to the PF.

You don't need curved spacetime for this. The classical Law of Gravitation works just fine here. Are you familiar with that equation? If not, try Wikipedia and let us know if you still have questions.
Thanks Berkeman, yes I am aware of the universal law of gravity (actually I just wrote a little paper on the equation and other motion-related, relatively straight-forward Newtonian equations for a high-school project.)

It's more that I'm trying to intuitively understand how this phenomenon works according to EINSTEIN'S gravity because I can't follow the complicated maths which would presumably allow you to follow and intuit or understand the work mathematically.

#### russ_watters

Mentor
It's more that I'm trying to intuitively understand how this phenomenon works according to EINSTEIN'S gravity because I can't follow the complicated maths which would presumably allow you to follow and intuit or understand the work mathematically.
Fortunately, for stationary point sources, the math is identical!

What you said in your first post doesn't make sense: If the objects are released from stationary near each other, they will accelerate toward each other, not remain motionless. Perhaps you are misunderstanding the concept of "proper acceleration" in GR. Proper acceleration is measured vs freefall (it is what an accelerometer measures), but that doesn't mean the objects are at rest with respect to each other: they are in freefall.

#### Orodruin

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They get closer because they are following what amounts to straight lines in spacetime - but spacetime itself is curved. The effect is similar to why straight lines on a sphere eventually meet.

#### Ibix

Have you drawn displacement-time graphs before? They're just graphs of the x-position of something as a function of time. One interpretation of the maths of special relativity is that you should take such graphs pretty much literally - time is a direction in spacetime just like it's a direction on your piece of paper (well, not "just like" - the time dimension is different from the spatial ones, but we don't need to care about that for this analogy). The implication of this is that a point object in space is actually a line in spacetime (and a non-point object is a broad line). If the object is not moving, it's just a straight line parallel to the time axis. If it is moving it's a sloped line, and if it's accelerating it's a curved line.

Newton's first law of motion says that objects move in a straight line unless acted on by a force. On a displacement-time graph this means that, unless there are forces, all objects are straight lines and objects moving at the same speed are parallel lines. However, there's a hidden assumption here - that the graph paper is flat. If the graph paper isn't flat (and this is where special relativity morphs into general relativity) it may not even be possible to draw a straight line. Orodruin gave the example of a sphere. If you and a friend start out moving parallel to each other (due north, for example), you'll find that your separation varies until you actually bump into each other at the pole. This is because the idea of "just going in the same direction", which produces a straight line on a flat piece of paper, produces a great circle on a spherical surface. Now spacetime isn't a sphere; it actually follows a class of geometry called pseudo-Riemannian geometry, with the specifics determined by the distribution of mass and energy. But the principle is the same - two objects using their own notion of "just going straight on in the same direction" doesn't necessarily lead to parallel lines.

So, in summary, the reason that objects move relative to one another in a gravitational field is a modified version of Newton's first law. Objects follow the nearest analogue to a straight line path that there is in the curved spacetime in which they reside ("follow geodesic paths") unless acted on by a force. And geodesics that are initially parallel aren't necessarily always parallel.

Hope that helps. There's a nice animation that one of our members put together that illustrates this. One day I'll remember to bookmark it, but for now I will just tag @A.T. and hope he adds it.

#### pervect

Staff Emeritus
Why do bodies in free fall change position relative to each other?

For instance in a vacuum with no external forces why can an apple fall towards the surface of the earth? If these objects aren't accelerating then are they considered at rest? And in what sense of the term 'rest'.

I think I can intuit how in a curved space time, WHEN a body experiences a force its movements are dictated by the curvature of the spacial geometry it moves through. I think my problem is I am only intuiting a curved space, rather than a curved space time.... Is it even possible to intuitively understand GR?

(To clarify: I'm just asking for help with the primary question, the rest was to attempt to give context to my thought to help reveal the issue in my understanding)
If a space-time graph has some intuitive meaning for you, you can consider a curved 2d spatial surface that represents a space-time with 1 time and 1 space dimension, a 1+1 space-time. Then you can understand the effects of curved space-time by imagining that you draw a space-time diagram on a curved surface, rather than a flat one.

The potentially tricky part is that you'll probaly intuit a space-time geodesic as a spatial geodesic on the 2d surface. This probalby isn't quite right, but it's close enough to get some vague idea of how it all works.

#### PeroK

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Thanks Berkeman, yes I am aware of the universal law of gravity (actually I just wrote a little paper on the equation and other motion-related, relatively straight-forward Newtonian equations for a high-school project.)

It's more that I'm trying to intuitively understand how this phenomenon works according to EINSTEIN'S gravity because I can't follow the complicated maths which would presumably allow you to follow and intuit or understand the work mathematically.
Let me first re-phrase your question. If there is no force in GR, why should a particle follow one path rather than another?

If you go back to Newtonian Mechanics, you can look at things using forces. But, you can also look at it using the Lagrangian principle. Basically, Lagrange found an equivalent formulation of mechanics (equivalent to Newton's laws) which was based not on forces but on the principle that nature operates in order to minimise (or maximise) a quantity called the Lagrangian.

Now, when you move to GR the forces disappear, but the Lagrangian principle remains. This is essentially a postulate of GR: that particles move in order to maximise the amount of "proper" time they experience. Proper time being the Lagrangian in this case. This leads, by the miracle of mathematics, to the same equations of motion (more or less) as particles subjected to a gravitational force.

#### Ibix

There's a nice animation that one of our members put together that illustrates this.
Found it!

#### A.T.

For instance in a vacuum with no external forces why can an apple fall towards the surface of the earth?
See video in post #9 and the one below:

"Why do bodies in free fall change position"

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