In curved space-time, what causes gravitational acceleration?

[ft_c]

Hi,
Just a quick question I can't get my head around... Say I have a big planet sitting in intergalactic space and I place myself somewhere in the vicinity, relatively stationary to it. So I am not moving, the planet is not moving, how do I then 'follow' a curve in space-time? Am I assumed to be already moving? Is space-time going somewhere? If so, how?!
Hope you can help!

Thank you!

 

[Dale]

Am I assumed to be already moving?

In a sense, yes. You are "moving" in the time direction, i.e. towards the future. Remember, spacetime is not just space, but space and time together.

I put "moving" in quotes because you have to be careful how you define things, however if you know some Riemannian geometry you can make the statement rigorous in terms of coordinate bases and tangent vectors.

 

[ghwellsjr]

Unless you place yourself on the surface of the big planet, you're not going to stay relatively stationary to it. You're going to fall onto the surface and it's going to hurt--bad.

 

[ft_c]

In a sense, yes. You are "moving" in the time direction, i.e. towards the future. Remember, spacetime is not just space, but space and time together.

Thanks DaleSpam!
Sorry to respond to your answer with more questions!
So are you saying that it is just...pre-mapped... for lack of better terminology!?
That if I am at 'A' at one point in time, then x amount of time later I will be at 'B'?

I haven't started my degree yet (my first module starts in 2 weeks!) so am clueless about the tensors and all that at the moment! But it seems as if a force is missing, or hasn't been defined - I can't see why that just because space-time is curved, I will be in a different position and be accelerating at a different rate at a different time. For instance, we have the example of a bowling ball on a trampoline and billiard balls rolling round the outside to describe solar systems etc, but in that example where does that downward force come from? It's using gravity to explain gravity!

Surely the particles in my body need to interact with space-time somehow in order to generate acceleration?

 

[DrGreg]

The "bowling ball on a trampoline" analogy isn't very good -- for the reasons you've given, and because it illustrates only curved space, not curved spacetime. A better picture is here: www.relativitet.se/spacetime1.html. Free-falling objects follow the straightest route they can through spacetime, which isn't necessarily a straight route through space.

 

[elfmotat]

But it seems as if a force is missing, or hasn't been defined - I can't see why that just because space-time is curved, I will be in a different position and be accelerating at a different rate at a different time.

In flat spacetime, objects travel in straight lines. If an object is stationary in a particular reference frame, then you can represent this with a simple vertical worldline on a spacetime diagram.

Similarly, objects follow "straight" lines through a curved spacetime, called geodesics. Picture the surface of a sphere: if you draw two "straight" lines that start out parallel, then eventually they intersect. Now imagine the surface of a sphere is a spacetime diagram representing two objects initially at rest with respect to each other (their worldlines are initially parallel). As you move along the lines, the distance between them decreases until they eventually intersect. This is similar to the way things work in relativity, but the curved spacetimes are more complex than that of the surface of a sphere.

EDIT: I see DrGreg beat me to it.

For instance, we have the example of a bowling ball on a trampoline and billiard balls rolling round the outside to describe solar systems etc, but in that example where does that downward force come from? It's using gravity to explain gravity!

The people around here tend to think that this analogy is a pretty bad one.

 

[Dale]

For instance, we have the example of a bowling ball on a trampoline and billiard balls rolling round the outside to describe solar systems etc, but in that example where does that downward force come from?

Yes, it is not a well-liked illustration here for several reasons. Let me explain in a less graphic but more relevant way.

I assume that you have done position-time diagrams, these are almost equivalent to spacetime diagrams, with a few traditional differences. Position-time diagrams traditionally use SI units with time plotted horizontally and position plotted vertically, while spacetime diagrams traditionally use units where c=1 with position plotted horizontally and time plotted vertically.

Ignore gravity for a moment. Suppose we plot the worldline for an inertially moving particle in a spacetime diagram. It winds up being a straight line. It is a vertical line if it is at rest, and it is at a ±45° angle if it is moving at c, and it is some other slope if it is moving with some v<c. Now, suppose we put an accelerometer on this particle, the accelerometer reads 0. So, inertially moving particles have 0 proper acceleration and are straight lines.

One important thing about straight lines in flat spaces is that if they are parallel then they never intersect. So now imagine a spacetime diagram with two parallel lines, as time goes on both particles always have 0 proper acceleration and they are always at rest wrt each other so the distance between them is always constant and they never collide.

Now, consider gravity physically for a moment, two objects interacting gravitationally will have accelerometers that read 0, but even if they are initially at rest wrt each other at one point in time the distance between them will change and they will eventually collide. So, in terms of worldlines in spacetime, we have two straight lines which are initially parallel, but eventually they intersect. This is impossible in flat spaces, but in curved spaces it happens.

Consider two nearby longitude lines on a sphere. Each one is "straight" (the generalization of "straight" to curved spaces is called "geodesic"), they start out initially parallel at the equator, but they intersect at the pole, despite never turning either left or right. If you consider North to be time and East to be space then this is a rough model of gravity.

 

[ft_c]

I see! This makes a lot of sense, thanks for all your answers.

Now, consider gravity physically for a moment, two objects interacting gravitationally will have accelerometers that read 0, but even if they are initially at rest wrt each other at one point in time the distance between them will change and they will eventually collide. So, in terms of worldlines in spacetime, we have two straight lines which are initially parallel, but eventually they intersect. This is impossible in flat spaces, but in curved spaces it happens.

This is really great :) thank you! It does beg the question, how might one of these particles curve spacetime?! I guess we don't know this...? is this where stuff like the Higgs comes in? Anyhow, on a slight tangent, is the possibility of an atemporal universe in this case then incompatible with relativity? i.e: is time (as we currently understand it) necessary, or could we still get along with relativity by thinking atemporally and replacing the word time with, for example, the 'medium of change'? (I had been thinking on these lines then found this essay: http://fqxi.org/data/essay-contest-files/Rovelli_Time.pdf )

 

[elfmotat]

This is really great :) thank you! It does beg the question, how might one of these particles curve spacetime?! I guess we don't know this...? is this where stuff like the Higgs comes in?

The curvature of spacetime is related to mass/energy/momentum by the Einstein Field Equations.

Anyhow, on a slight tangent, is the possibility of an atemporal universe in this case then incompatible with relativity? i.e: is time (as we currently understand it) necessary,

I'm not really sure what you're asking. Time is one of the four dimensions in GR, and without it the theory wouldn't work.

or could we still get along with relativity by thinking atemporally and replacing the word time with, for example, the 'medium of change'?

What's the difference between time and "medium of change?" How does changing the word make it any different?

 

[Dale]

It does beg the question, how might one of these particles curve spacetime?!

I agree with elfmotat's comments above, that is exactly what the Einstein Field Equations describe. However, you really need some background in tensors and pseudo-Riemannian geometry before you are ready to tackle that.

could we still get along with relativity by thinking atemporally and replacing the word time with, for example, the 'medium of change'?

You cannot get rid of time simply by making the substitution "time"->"medium of change" any more than you could get rid of sickness by making the substitution "sickness"->"illness".

 

[ft_c]

The curvature of spacetime is related to mass/energy/momentum by the Einstein Field Equations.

I understand that curvature is related to mass/energy/momentum, and this is great, we have the means to figure out the topology (right word?) of spacetime based on these factors, but what mechanism is actually being described?? How do mass/energy/momentum affect curvature and why??

I'm not really sure what you're asking. Time is one of the four dimensions in GR, and without it the theory wouldn't work.

What's the difference between time and "medium of change?" How does changing the word make it any different?

Yes it is a bit weird to imagine an atemporal universe, and 'medium of change' is a terrible choice of words to use! But I am under the impression that matter is trapped energy, in some form - I imagine it as electromagnetic radiation.

EM radiation traveling at the speed of light experiences no time, hence if matter is a bunch of EM radiation it also does not. So creatures (us humans) existing in the system (the universe) as a product of that system, experience time only as the progression of bits of matter, of which bits themselves experience no time, but as a result of clockwork, years/days/seconds, memory - chemicals whizzing around the brain, etc... the creatures can measure 'time', and use it evolutionarily as a gigantic advantage.

For many reasons I imagine time as an extremely useful construct of instinct, but feel as if (dare I say?!) it may hold back progress of science towards the ultimate goal of a theory of everything (eek!). I initially came to this by thinking quite amateurishly about Einstein's relativity and my own picture of elementary particles, and that there is only one speed for anything in the universe, nothing ever goes any faster or slower than the speed of light - and so, is time as we perceive it really a necessary quality of the universe? With my new understanding of gravity under relativity (thank you guys for that very much!) I wondered if you would have any thoughts on whether this view of 'time' would be incompatible with relativity with regards to gravity?

 

[Dale]

what mechanism is actually being described?? How do mass/energy/momentum affect curvature and why??

How they affect curvature is completely given by the EFE, that is the mechanism.

Why is a different question and the answer really depends on what you are allowed to assume as a given:


GR is currently the fundamental theory of gravity. We can accept GR as a given and explain "why Newtonian gravity" as an approximation to GR, but we currently do not have another theory which we can accept as a given and use to explain GR. And even if we did you could just switch your "why" question to that theory and then you would be in the same situation.

There will always be some fundamental theory which cannot be explained in terms of other theories (which is what it means to be fundamental). Those theories stand by themselves on the basis of their ability to predict the results of experiments. If they predict correctly then they are accepted, if they do not then they are rejected.

But I am under the impression that matter is trapped energy, in some form - I imagine it as electromagnetic radiation.

This site is for mainstream science, not personal speculation. EM radiation does not have charge, it has integer spin, and it does not participate in either the strong or the weak interaction.

 

[ft_c]

This site is for mainstream science, not personal speculation. EM radiation does not have charge, it has integer spin, and it does not participate in either the strong or the weak interaction.

I absolutely agree - I assure you my speculations are based on as much currently understood science as I can possibly take in at the moment! I have seen that whole Feynman interview, and it is for the reasons he talks about why I am going to study my physics BSc :)

Of course photons are chargeless, but atoms are colourless and I've got a red pen! As in Feynman's diagrams, electrons coming into close proximity emit and receive photons towards each other, conservation of momentum etc. - they repel. So I've been trying to work out a system to explain where those photons are coming from and how they know where to go. I think the strong force is not directly related to EMR, and I thought the weak force was completely governed by electromagnetism??

When I get round to learning the EFE's, do you mean I will get to learn the cause of spacetime curvature? Or only the effect? If we knew the cause we surely wouldn't be looking for the Higgs (which incidentally I don't think we will find!)

This is really what I meant by the 'how'!

Anyway to go back to my previous question, would this atemporal view of the universe have any inconsistencies with GR? To plot a graph of space (space with no time) would still absolutely require another dimension to account for motion, or progression of events, so this makes me think it would still be compatible. I think a better question would be, is it definitely time that is required for GR to work?

 

[Dale]

my speculations are based on as much currently understood science as I can possibly take in at the moment!

Since you realize that you have not taken in all of the currently understood physics, you would be best served to fill in that gap in your own knowledge before seeking to fill in the gaps in the field as a whole. If you start speculating from a place of ignorance then:
1) often you concentrate on non-problems or problems which have already been solved
2) you generally find obviously unworkable solutions which have already been discarded
3) if you find workable solutions they have generally been found already

Furthermore, even once you have learned it all, if you speculate on this forum:
1) you get banned

I am not going to engage you in a discussion about your speculative ideas, I am not interested in them myself nor do I wish to get banned.

Of course photons are chargeless, but atoms are colourless and I've got a red pen!

It most certainly is not true that atoms are colourless. In fact, spectroscopy is the most important tool for determining which atoms exist in a given stellar object.

When I get round to learning the EFE's, do you mean I will get to learn the cause of spacetime curvature?

Yes, according to the EFE the cause of spacetime curvature is the stress-energy tensor. Although I am unsure if you consider cause and cause to be different concepts.

 

[ft_c]

It most certainly is not true that atoms are colourless. In fact, spectroscopy is the most important tool for determining which atoms exist in a given stellar object.

Atoms do not possesses colour when not excited though, they emit photons of a colour only when excited by that colour. If you are saying colour is a similar property to charge, then you are saying that when an electron is not being excited by some other electric field it doesn't possesses electrical charge.

 

[Dale]

Atoms do not possesses colour when not excited though, they emit photons of a colour only when excited by that colour.

By that same rationale your red pen also does not "possess colour" when in a dark room.

If you are saying colour is a similar property to charge

I certainly never made that analogy; I think it is a pretty weak analogy. I was merely correcting a counter-factual statement on your part about atoms being colorless.

 

[ft_c]

Yes exactly, my pen is not red when the light is off! So this was my initial point - photons don't carry a charge in the same way atoms don't carry colour...

 

[ft_c]

Hang on - another thought here! If you go really fast, time slows down, so we're moving slower in the time direction - then there would be less gravity as you're not progressing through the curvature of space as fast as usual?? There must then be a speed at which you can go at which you don't experience any gravity at all? Light doesn't experience time as it travels at c, so how is it at all affected by the curvature of space-time? Is this a problem of FoRs?

 

[Dale]

Yes exactly, my pen is not red when the light is off!

So there is nothing different between atoms and pens wrt color. Both pens and atoms have color when illuminated and do not when not illuminated. Your initial assertion that the pen has color and the atoms are colorless is incorrect.

 

[ft_c]

Yes yes ok you win! I know what I meant anyway... :P

 

[Dale]

Hang on - another thought here! If you go really fast, time slows down, so we're moving slower in the time direction - then there would be less gravity as you're not progressing through the curvature of space as fast as usual?? There must then be a speed at which you can go at which you don't experience any gravity at all? Light doesn't experience time as it travels at c, so how is it at all affected by the curvature of space-time? Is this a problem of FoRs?

The reason that GR is formulated in terms of tensors and pseudo-Riemannian geometry is that it allows you to express the laws of gravity and physics in terms that are completely independent of the frame of reference, and also to calculate quantities that are independent of the frame of reference. Because of that, all of these invariant quantities are the same if you are a small cosmic ray whipping past the Earth or if you are a stationary atom next to a super-relativistic planet.

 

[harrylin]

Hang on - another thought here! If you go really fast, time slows down, so we're moving slower in the time direction - then there would be less gravity as you're not progressing through the curvature of space as fast as usual?? There must then be a speed at which you can go at which you don't experience any gravity at all? Light doesn't experience time as it travels at c, so how is it at all affected by the curvature of space-time? Is this a problem of FoRs?

There is an effect somewhat similar to electromagnetism, so that the effective gravitation is a function of speed. However, if I'm not mistaken the effect works not to decrease but to increase the bending up to a factor of two at the speed of light for motion tangential to the field (if not, I'll be happily corrected). Anyway, this is definitely not a problem of FoRs, as light doesn't have one.

The way light is affected in terms of ordinary geometry was first(?) clearly described in 1916, here, on p.198-199 of the English translation:
http://www.Alberteinstein.info/gallery/gtext3.html (watch out, even the lesser resolution is a big file!).

 

[pervect]

There is an effect somewhat similar to electromagnetism, so that the effective gravitation is a function of speed. However, if I'm not mistaken the effect works not to decrease but to increase the bending up to a factor of two at the speed of light for motion tangential to the field (if not, I'll be happily corrected). Anyway, this is definitely not a problem of FoRs, as light doesn't have one.

The way light is affected in terms of ordinary geometry was first(?) clearly described in 1916, here, on p.198-199 of the English translation:
http://www.Alberteinstein.info/gallery/gtext3.html (watch out, even the lesser resolution is a big file!).

The rigorous way of describing the path light takes is to say it follows a null geodesic. The geodesic equations for this motion can be written down and solved, the process is the same for light (null geodesics) and matter (timelike geodesics), only the initial conditions vary.

There isn't any truly rigorous way of describing the motion of light in gravity accurately in terms of "forces" that I'm aware of that has been published.

One can make various attempts, but one runs into various confusions over the concept of what "force" means in the context of a curved space time that the texts don't really address. The texts DO tell you how to work mathematically with curved space-time, but they DON"T tell you how to unambiguously define forces in curved space-time, at least not the basic texts.

Alas, many of the questions we get from the lay audience tend to be pre-cast in the mold of forces - a mold that doesn't really fit the problem.

The case of the "extra" bending of light is a good example. The mathematics is clear, but in words the issue is fuzzy. My best attempt would be to attribute the extra bending of light to the curvature of space. (Not space-time, but space). This is something that is not exactly a "force" as we are used to thinking of forces.

Is something that's traveling a straight line (i.e. a spatial geodesic) in a curved space really subject to any sort of forces? I would say no, it's really just traveling in a straight line. Note that the actual path it follows is independent of its speed. But if you go to calculate light deflections in terms of coordinates, the spatial curvature does affect how fast the coordinates change, i.e. the second derivative of the rate of change of the spatial coordinates.

If you ignore all the effects caused by curved space, you can describe the rest of the motion as being due to "forces" easily enough, but you wind up being off by a factor of two because you've neglected the effects of spatial curvature.

You might also try to say (and I've see amateur posters do this, though never professionallly published books or papers) that the second derivative of some particular coordinate in some particular coordinate system represents an adequate definition of "force", using the f=ma analogy. The problem with this is that this trick doesn't provide a way to think of the physics independent of the coordinate system - you can't really work any problem this way without completely specifying the coordinates you are using, trying to define force in this manner makes the concept of what you mean by a "force" coordinate dependent.

 

[harrylin]

The rigorous way of describing the path light takes is to say it follows a null geodesic. The geodesic equations for this motion can be written down and solved, the process is the same for light (null geodesics) and matter (timelike geodesics), only the initial conditions vary.

There isn't any truly rigorous way of describing the motion of light in gravity accurately in terms of "forces" that I'm aware of that has been published.
[..]
The case of the "extra" bending of light is a good example. The mathematics is clear, but in words the issue is fuzzy. My best attempt would be to attribute the extra bending of light to the curvature of space. (Not space-time, but space). This is something that is not exactly a "force" as we are used to thinking of forces. [...]

Thanks for the precisions. I had overlooked that not the OP was trying to describe the motion of light in gravity in terms of "forces". However, if he reads the pages that I referred to, he may notice that in GR light is typically modeled as a wave and the bending is calculated without the introduction of forces. I think that we may say that in GR the experience of gravitational force is an effect and not a cause.