# If a body is initially stationary, it falls when released. Why does it fall along a geodesic?

Ibix
quote: "these transforms could be interpreted as meaning that space and time were part of one 4d structure, christened spacetime." Would anyone care to expand on this a bit, please?
It's difficult without knowing how much maths you know. Riemann and others had done a lot of work on the geometry of smooth spaces - more general spaces than the Euclidean plane geometry you probably studied in school. Minkowski apparently recognised that the Lorentz transforms were the same maths that represents "rotations" (typically called boosts, but they are the hyperbolic analogue of rotations) in a type of space now named after him. I don't know if that space was known before Minkowski or if it was his own personal invention. Either way, it's a four dimensional "space" with one dimension that's different from the other three and whose behaviour under boosts matches Einstein's maths. That's spacetime.

Note that I believe that other interpretations of the maths are possible. But using geometrical language opens up the whole toolkit of Riemann's differential geometry, so everyone uses that.

Klystron and phinds
They are geodesics in spacetime. You can't not be moving forwards in time.

You are probably being misled by pictures showing space around Earth as curved. In such diagrams, all the interesting stuff happens in the time direction - which isn't shown!
What I now think I know is that as the earth moves through spacetime it bends spacetime as it travels along. When earth moves along a bit it leaves the bent piece of spacetime behind and it straightens itself out as the earth bends a new piece as it were. It's a dynamic process. So in fact my mass cannot be stationary relative to spacetime. It is impossible. I must admit I had a different image of how it happens all in my mind until the penny dropped. I imagined that the particular piece of spacetime which the earth bent actually stayed with the earth. Of course this image was wrong, I see that now. And another mass that is stationary with respect to the centre of the earth but is in the flow of the earth's bent spacetime must cut the geodesics and is therefore accelerating and therefore feels a force, the one we call gravity. Have I got this right?

PeroK
It's difficult without knowing how much maths you know. Riemann and others had done a lot of work on the geometry of smooth spaces - more general spaces than the Euclidean plane geometry you probably studied in school. Minkowski apparently recognised that the Lorentz transforms were the same maths that represents "rotations" (typically called boosts, but they are the hyperbolic analogue of rotations) in a type of space now named after him. I don't know if that space was known before Minkowski or if it was his own personal invention. Either way, it's a four dimensional "space" with one dimension that's different from the other three and whose behaviour under boosts matches Einstein's maths. That's spacetime.

Note that I believe that other interpretations of the maths are possible. But using geometrical language opens up the whole toolkit of Riemann's differential geometry, so everyone uses that.
Thank you IBIX. I do know a bit of maths as it happens so now I have a link which I can follow up.

I'll think about it. Thanks for now.
I've thought about it now and asked several more questions and every time I've received helpful replies. I want to thank everyone for their help (and forbearance against my stupidity). You have all been great.

Don't think, @woolyhead77, that this stuff came naturally to anyone. The maths of special relativity (published 1905) is implicit in Maxwell's equations (published around 1862). It took forty years to puzzle out, and another ten to expand special relativity to general relativity. And there are still arguments about how best to teach it and it can take years to get your head around it.

But the evidence is compelling. Without GR we can't explain the behaviour of light near masses, nor the exact precession of Mercury' orbit, nor the behaviour of clocks at different heights, nor the redshift of distant galaxies... "Get used to it" is a fairly blunt way of putting it, but an awful lot of "common sense" knowledge about the universe is hilariously wrong outside the limited range of our everyday experience.
Thank you. I appreciate your kindness in the way you answer. And I am getting used to the idea of spacetime after receiving a lot of help from everyone.

PeroK
Homework Helper
Gold Member
What I now think I know is that as the earth moves through spacetime it bends spacetime as it travels along. When earth moves along a bit it leaves the bent piece of spacetime behind and it straightens itself out as the earth bends a new piece as it were. It's a dynamic process. So in fact my mass cannot be stationary relative to spacetime. It is impossible. I must admit I had a different image of how it happens all in my mind until the penny dropped. I imagined that the particular piece of spacetime which the earth bent actually stayed with the earth. Of course this image was wrong, I see that now. And another mass that is stationary with respect to the centre of the earth but is in the flow of the earth's bent spacetime must cut the geodesics and is therefore accelerating and therefore feels a force, the one we call gravity. Have I got this right?
Note that in curved spacetime and the theory of GR there are no preferred reference frames. It's not really possible to say that this view of spacetime with the Earth moving is correct and a different view with the Earth at a fixed point in space is not. In fact, if you study the spacetime around a large object like the Sun or the Earth, it's usual to look at this as a static scenario. I.e. you can describe the spacetime in such a way that it does not change with your time coordinate.

The Earth itself is following a geodesic as it orbits the Sun, which can be seen as a static field. If you include the Earth (and other planets) in your model as objects large enough to affect the spacetime in the Solar system, then you get a dynamic solution, which changes as the planets change their positions relative to the Sun and each other. But, there is nothing absolute about this view.

For an object on the surface of the Earth, a simple view is that the geodesic is towards the centre of the Earth, and it's the force from the Earth's surface that prevents the object following this. This isn't the force we call gravity, which would be a force towards the centre of the Earth. The force we call gravity is the fictitious force that counteracts this upward force. In Newtonian physics we must have this force of gravity so that forces are balanced.

In GR, the object has a real upwards force and a proper acceleration but owing to the curvature of spacetime the surface of the Earth at that point represents an accelerating reference frame. The net result is that the object remains on the surface.

If you look at that object and ask "if that object has an upwards force, why isn't it accelerating relative to me?", then the answer is that you too have the same proper acceleration.

If you look at an object falling "under gravity" and ask "why is it accelerating, when it has no force on it?", then the answer again is because you have proper acceleration.

PeterDonis
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2019 Award
as the earth moves through spacetime it bends spacetime as it travels along. When earth moves along a bit it leaves the bent piece of spacetime behind and it straightens itself out as the earth bends a new piece as it were
This is not really a correct description, although it has some heuristic value.

The earth does not "move through spacetime". In spacetime, the "earth" is a "world tube"--a tube that occupies a certain portion of spacetime. Spacetime is curved near this world tube because, as you say, the earth causes spacetime curvature--or, to put it another way, there is stress-energy present inside this world tube, which causes the spacetime in and near it to be curved. But this world tube does not "move"--it simply is.. It is a geometric object, a tube, lying within a larger geometric object, all of 4-d spacetime. The Earth, or more precisely its stress-energy, is "there" in every part of that world tube. It does not "move" from one part to another.

At the center of the Earth's world tube is one particular worldline, the worldline of its center of mass. This worldline is a geodesic; if we take a larger scale view we see that it is a geodesic in the curved spacetime surrounding the Sun (which occupies a larger world tube distant from that of the Earth).

A person standing on the surface of the Earth is described by a worldline (or a world tube with a much, much smaller diameter) that runs parallel to the Earth's world tube and just touching its boundary. An object freely falling in the Earth's vicinity is described by another worldline (or world tube with small diameter) that is not parallel to the Earth's world tube. The latter worldline is a geodesic. There are also other geodesics, much farther from Earth, that are much closer to being parallel to the Earth's world tube, because spacetime out there is much closer to being flat.

Motore and PeroK
phinds
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2019 Award
A person standing on the surface of the Earth is described by a worldline (or a world tube with a much, much smaller diameter) that runs parallel to the Earth's world tube and just touching its boundary.
I'm confused. Wouldn't the person's tube be spiraling around the Earth's tube, since the Earth is rotating on its axis?

PeterDonis
Mentor
2019 Award
Wouldn't the person's tube be spiraling around the Earth's tube, since the Earth is rotating on its axis?
Yes, that's correct. I was thinking of an idealized non-rotating planet.

phinds
quote: "these transforms could be interpreted as meaning that space and time were part of one 4d structure, christened spacetime." Would anyone care to expand on this a bit, please?
Oh well, never mind. It sounds like a stupid question anyhow. What I should have said is can someone lead me through the process. But I'm sure you must be ready to tear your hair our with me so I'll thank you all and let the matter drop.

PeroK
Homework Helper
Gold Member
Oh well, never mind. It sounds like a stupid question anyhow. What I should have said is can someone lead me through the process. But I'm sure you must be ready to tear your hair our with me so I'll thank you all and let the matter drop.
It's not a stupid question, but it is a big subject. You could start here:

https://en.wikipedia.org/wiki/Minkowski_space

Ibix
Ibix
Oh well, never mind. It sounds like a stupid question anyhow. What I should have said is can someone lead me through the process. But I'm sure you must be ready to tear your hair our with me so I'll thank you all and let the matter drop.
It's either far too complicated to go into or I've more or less done it.

Basically, Einstein realised that Lorentz' ad hoc patch to Maxwell's equations could be used to derive a revision of basic kinematics that looks really like Newtonian physics as long as nothing is moving too fast (which is why we didn't notice until we began to look into electromagnetism - even a rifle bullet is too slow by this standard). Minkowski, as far as I know, simply recognised that Einstein's maths looked like Riemann's maths describing a (3+1) dimensional space, and said so. The idea turned out to have truly useful theoretical legs because Newtonian gravity cannot work in relativity (nothing can travel faster than light, but Newtonian gravity is supposed to be infinitely fast), and trying to make special relativity work in curved coordinates (rotating reference frames) led people (notably Einstein) to the idea that gravity might be modelled as a curved version of Minkowski's spacetime.

If you want a lot more detail than that you're going to need a book. I usually reccomend Taylor and Wheeler's Spacetime Physics to people wanting to learn special relativity, but you seem to be more interested in the historical development than the physics. I'd try PeroK's link. If not, somebody posted a reference on the history of relativity the last time I wrote all this out - unfortunately I recall neither the reference nor the poster. Hopefully they'll post again, and hopefully I'll make a note this time.

bhobba
Mentor
And there are still arguments about how best to teach it and it can take years to get your head around it.
Most definitely. Ohanian in his book Gravitation and Space-Time does it by first noting a theorem by Wigner that says all fields must be Tensors. Electromagnetism is based on a field with a tensor Au. Ok what would be the next most simple field - why Φuv of course (excluding the simplest of all - a scalar). So build a theory based in a similar way to EM but out of Φuv instad Au. It's not hard - in the Lorentz gauge (∂uAu=0) we have ∂v∂v Au = Ju so we assume in a similar gauge (∂uΦuv=0 - called the Hilbert gauge) and ∂w∂w Φuv = Tuv where Tuv is called the stress-energy tensor. Analyse it, and wonder of wonders it behaves like a relativistic extension of Newtonian gravity - and, being sneaky, what we define as the stress energy tensor turns out to act like the stress-energy tensor of SR thus justifying the name. Because of this its called linearised gravity. Then we do a bit more analysis and find it makes space-time act like it has an infinitesimal curvature, and its gauge invariance is just infinitesimal coordinate transformations. Very strange. Then we naturally ask what if space-time is more than just infinitesimally curved and is invariant to all coordinate transformations not just infinitesimal ones (which of course would be natural for a geometrical view - the geometry of course should not depend on coordinate transformations). Then something equally as strange - perhaps even a miracle - happens - your hands are tied - you get Einsteins Field Equations. So in this approach space-time curvature emerges by assuming a flat space-time and following the math. It was the first serious GR book I read - then I read Wald which is at a more advanced level and very geometrical. Then I read the famous MTW which is a bit more intuitive.

Thanks
Bill

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