Why do we fall, according to GR?

[kweba]

I've been searching around this forum and the internet, and I still have a hard time understanding how curved spacetime, as described by General Relativity, can cause us and objects to fall back on the surface of the Earth. I get the concept of Geodesics and how planets revolve in orbit around the sun (because of curved straight paths). But I don't really get how GR describes why we fall.

Is it true spacetime push as back to Earth?

Thanks in advance to those who could help me clear this up.

 

[Muphrid]

Objects that fall simply follow geodesics that lead back to the Earth. Gravity influences the shapes of geodesics in any region.

 

[Jonathan Scott]

I've been searching around this forum and the internet, and I still have a hard time understanding how curved spacetime, as described by General Relativity, can cause us and objects to fall back on the surface of the Earth. I get the concept of Geodesics and how planets revolve in orbit around the sun (because of curved straight paths). But I don't really get how GR describes why we fall.

Is it true spacetime push as back to Earth?

Thanks in advance to those who could help me clear this up.

The important thing is to note that it is the curvature of spacetime, not just space, which gives rise to gravity. For example, if you take an object at rest and plot its radial distance from a gravitational source against time on graph paper, you will get a line which curves towards the gravitational source. This effectively shows the curvature of the path (relative to a static coordinate system) with respect to time. The curvature is the same as the acceleration expressed in units where c=1.

In simple central source situations, the curvature of space (relative to the coordinate system) in GR is the same as that with respect to time, but this is barely measurable. The curvature in ordinary units is g/c² which means the radius of curvature is c²/g. For the typical gravitational field of the earth, the radius of curvature works out to about a light year.

The curvature with respect to time is much more noticeable because we are effectively "moving through time" with speed c.

Planets do not move on their orbits because of the curvature of space. Their orbits are primarily determined by the curvature with respect to time, as for all other objects moving at non-relativistic speeds. It is only when something is moving at or near the speed of light (such as light rays or radio waves passing close to the run) that the curvature with respect to space also has a significant effect.

 

[Naty1]

Their orbits are primarily determined by the curvature with respect to time, as for all other objects moving at non-relativistic speeds...The curvature with respect to time is much more noticeable because we are effectively "moving through time" with speed c.

yes.
Most all slow moving massive matter curves time more than space... it is NOT easy
to visualize...Another way to say what Scott posts is that the very high speed of 'c' time curvature swamps the effects of space curvature.

I still have no idea how time can 'curve' but it seems to describe observations [experimental results]. It's a bit like asking another guy "How do women think?" No male really knows.

 

[Nugatory]

I've been searching around this forum and the internet, and I still have a hard time understanding how curved spacetime, as described by General Relativity, can cause us and objects to fall back on the surface of the Earth. I get the concept of Geodesics and how planets revolve in orbit around the sun (because of curved straight paths). But I don't really get how GR describes why we fall.

Is it true spacetime push as back to Earth?

Thanks in advance to those who could help me clear this up.

In the GR view of the world, things do not fall, not in the sense of being pulled towards the surface of the earth. Instead they float happily along under their own inertia... while the surface of the Earth accelerates towards them at 10 m/sec².

And as for how curvature can produce this effect? Well, imagine that you and I are standing a few meters apart at the equator, and we start walking due north. After a few thousands of kilometers, we'll notice that we're drawing closer to one another, and by the time we reach the north pole, we'll collide. If we didn't know that the Earth was round, we'd think that some force was drawing us north-moving travellers towards each other.

Of course we travelers have to be moving to experience the effects of curvature. So what are we to make of an object that's at rest (maybe I'm holding it in my hand at the top of a tall building) and then released? Well, even an object that's at rest is moving forwards in time, so if space-time is curved in such a way that the natural paths through time of the object and of the surface of the Earth intersect, the object and the surface of the Earth will move towards one another and eventually collide - which is what "falling" is all about.

Before I drop the object, while I'm still holding it at the top of the building, my hand is pushing it upwards, exerting a force on it that pushes it off of its natural geodesic and inertial trajectory that will intersect the ground.

 

[DaveC426913]

I think the relevant question that might get to the bottom of the OP's curiosity is:

If an object were placed at a certain distance from a massive body like Earth such that it had zero initial velocity wrt Earth, why, in GR, does it begin falling toward the Earth?

 

[andrewkirk]

Oh that's a nice question for illustrating the curvature of spacetime DaveC. My guess - as a complete amateur - is as follows:

Graph the object's height above the Earth on the vertical axis against time on the horizontal axis. The object's initial four-velocity can be plotted on this graph as a horizontal vector of length 1 (with the two irrelevant spatial dimensions suppressed). So the object follows the unique geodesic on the graph that passes through the point (0,h) where h is the object's height above the Earth in metres, with tangent vector equal to the object's initial four-velocity (the horizontal line on the graph).

That geodesic gradually curves downwards towards the time axis, which represents the object's four-velocity gaining an increasing radial component in the initial reference frame.

 

[Dale]

I've been searching around this forum and the internet, and I still have a hard time understanding how curved spacetime, as described by General Relativity, can cause us and objects to fall back on the surface of the Earth. I get the concept of Geodesics and how planets revolve in orbit around the sun (because of curved straight paths). But I don't really get how GR describes why we fall.

Is it true spacetime push as back to Earth?

Thanks in advance to those who could help me clear this up.

I would say that the reason why we fall is the same in GR as in Newtonian mechanics: the force of gravity accelerates us towards the ground. The only difference is that in GR the force of gravity is a fictitious force, but in a reference frame where a fictitious force exists it still accelerates objects and does work and all of the other things you expect of forces.

 

[kweba]

To all of the repliers:

Thanks for the posts! I'm pretty sure I'm going to need to read through all of these again as the ideas have not still sink in my head. But from what I can understand so far, you guys are saying that time has something to do with it? I mean the bottomline is that it's because of time that we fall/accelerate back to Earth? That is bizarre! Haha but I get it and it make sense (I hope), since time is curved too, along with space.

But yes, the thing that I was really confused about is how an object at rest would follow that geodesic path down back to Earth, since it is not moving, unlike planets who are in orbit around the sun because of their linear inertial velocities that would follow the curved straight paths. The planets themselves are moving that's why it was easier for me to understand that concept, unlike an object at rest but would move along that geodesic and "fall".

So the answer is time?

 

[kweba]

Objects that fall simply follow geodesics that lead back to the Earth. Gravity influences the shapes of geodesics in any region.

Yes I understand that, but I'm talking about an object at rest. How would it follow and move along that geodesic if it's at rest?

 

[HallsofIvy]

It's not at rest in time, nothing is. Indeed, there is no such thing as "rest" in relativity.

 

[kweba]

The important thing is to note that it is the curvature of spacetime, not just space, which gives rise to gravity.

Yes thank you for the reminder, I understand it. I just took out "time" on purpose so that it could be simplified, and that it is harder to visualize time as another dimension. But I can see from what you're saying, time has something to do with the mechanics, and is an important factor. So pardon me.

Planets do not move on their orbits because of the curvature of space. Their orbits are primarily determined by the curvature with respect to time, as for all other objects moving at non-relativistic speeds. It is only when something is moving at or near the speed of light (such as light rays or radio waves passing close to the run) that the curvature with respect to space also has a significant effect.

(On another topic) So this explains time dilation for photons and light itself?

 

[kweba]

I think the relevant question that might get to the bottom of the OP's curiosity is:

If an object were placed at a certain distance from a massive body like Earth such that it had zero initial velocity wrt Earth, why, in GR, does it begin falling toward the Earth?

YES! Thank you. This is exactly what I'm confused about. An object at rest, but yet in motion through curved spacetime.

 

[yossell]

I would say that the reason why we fall is the same in GR as in Newtonian mechanics: the force of gravity accelerates us towards the ground. The only difference is that in GR the force of gravity is a fictitious force, but in a reference frame where a fictitious force exists it still accelerates objects and does work and all of the other things you expect of forces.

What's a fictitious force? How can a fictitious force exist? What's wrong with the standard GR explanation -- that in GR gravity is not a force, rather objects continue to move on geodesics, and mass/energy affects the shape of geodesics?

 

[DaveC426913]

I would say that the reason why we fall is the same in GR as in Newtonian mechanics: the force of gravity accelerates us towards the ground. The only difference is that in GR the force of gravity is a fictitious force, but in a reference frame where a fictitious force exists it still accelerates objects and does work and all of the other things you expect of forces.

Sorry, DaleSpam, but this doesn't answer the question at all. How does GR explain this fictitious force?

 

[Muphrid]

YES! Thank you. This is exactly what I'm confused about. An object at rest, but yet in motion through curved spacetime.

Picture a Cartesian coordinate grid. Over time, the grid distorts, being stretched and pulling inward toward the center of the Earth. GR tells us how much this warping happens based on the distribution of mass. The object at rest with respect to Earth is "moving" compared to the constantly stretching coordinate lines. When released by whatever outside force that held it up, it keeps its velocity at that momenr compared to coordinate lines, just as a free particle would. But the coordinate lines are accelerating toward Earth, so the particle falls.

This is an inexact visualization, but it should give an idea of what's going on.

 

[Dale]

What's a fictitious force? How can a fictitious force exist? What's wrong with the standard GR explanation -- that in GR gravity is not a force, rather objects continue to move on geodesics, and mass/energy affects the shape of geodesics?

It is the same explanation, just in more familiar terms.

A fictitious force is a force which exists because the coordinate system that you are using is not an inertial coordinate system. Classical examples are the centrifugal and Coriolis forces in a rotating reference frame. In the rotating reference frame these forces cause objects to accelerate and can do work etc. However, these forces are called fictitious because if you transform back into the inertial coordinate system then they disappear.

Similarly in GR. Gravity is a fictitious force. In the usual reference frame on the surface of the Earth it points down, but if you transform to a local free-falling frame then it will disappear. In the usual reference frame gravity pulls you down, like normal. In the free-falling reference frame the surface of the Earth accelerates upwards. In both cases, an object and the ground accelerate relative to one another.

 

[DaveC426913]

It is the same explanation, just in more familiar terms.

A fictitious force is a force which exists because the coordinate system that you are using is not an inertial coordinate system. Classical examples are the centrifugal and Coriolis forces in a rotating reference frame. In the rotating reference frame these forces cause objects to accelerate and can do work etc. However, these forces are called fictitious because if you transform back into the inertial coordinate system then they disappear.

Similarly in GR. Gravity is a fictitious force. In the usual reference frame on the surface of the Earth it points down, but if you transform to a local free-falling frame then it will disappear. In the usual reference frame gravity pulls you down, like normal. In the free-falling reference frame the surface of the Earth accelerates upwards. In both cases, an object and the ground accelerate relative to one another.

You're dancing around the issue!

Yes, I was going to use Coriolis force as an example.

So the OP knows that the Coriolis Force is fictional but wants to understand the best way to view it so that a fictional force is not invoked. We explain that on (or in) a rotating body, simple inertia will result in a straight path, while it is the observer that is curving.

So, how would you describe an object falling toward Earth without hand-waving it as a fictitious force? That doesn't explain what causes the movement. It's a cop out.

Why would the Earth accelerate upwards?

 

[Muphrid]

The surface of the Earth moves outward with respect to coordinate lines that are continually being sucked toward the center of the Earth.

 

[DrGreg]

This is exactly what I'm confused about. An object at rest, but yet in motion through curved spacetime.

A. Consider a freely-moving object when there is no gravity. We know such an object moves in a straight line at a constant velocity (relative to any inertial frame). If we draw a graph of distance against time on a flat piece of paper, we get a (red) straight line. Even if the object is at rest in the frame, we get a line parallel to the time axis, not a point.

B. Now consider the same freely-moving object with no gravity being observed by an accelerating observer. This can be represented by the same flat piece of paper as before with a red straight line on it, but now with curved blue grid lines instead of straight grid lines. If the red line starts off parallel to one of the curved blue gridlines, it won't remain parallel for long. In other words an object released from rest will appear to "fall" relative to the accelerating observer.

C. Finally, consider a freely-moving object falling under Earth's gravity. Now we need to draw our distance-against-time graph on a curved piece of paper. The object will now follow the straightest (red) line possible (a geodesic) on the curved sheet. The blue grid lines representing something at rest relative to the Earth do not follow the straightest routes on the sheet. If the red geodesic line starts off parallel to one of the blue non-geodesic gridlines, it won't remain parallel for long. In other words an object released from rest will fall.

 

[Dale]

You're dancing around the issue!

I am not dancing around the issue, I am just providing an equivalent alternative picture. Often one approach may not work for a particular individual and another will. The OP asked some follow-up questions which I answered, he will either decide the explanation doesn't work or he will continue to ask questions until he understands.

So, how would you describe an object falling toward Earth without hand-waving it as a fictitious force? That doesn't explain what causes the movement. It's a cop out.

It is not a cop out, it is a perfectly legitimate approach. It is completely equivalent to what you are saying. I don't know why you would say something like this, as though I am doing something dishonest in my reply.

If I wanted to avoid handwaving then I would associate the fictitious forces with the Christoffel symbols, as is usually done when you want to be rigorous. Then I would demonstrate how Christoffel symbols are zero in regular inertial reference frames in flat spacetime and non-zero in non-inertial reference frames even in flat spacetime. Then I would describe how in curved spacetime you can find a coordinate system where the Christoffel symbols are 0 at any given event in spacetime (local inertial frames), but not globally. In a local inertial frame there are no fictitious forces, and objects move only under the effect of real forces, but you can only do that locally. Globally in curved spacetime you will have to use non-inertial coordinates which mean non-zero Christoffel symbols and non-zero fictitious forces. In the weak field limit these fictitious forces are equal to the Newtonian force of gravity plus any Newtonian fictitious forces.

I would rather wave my hands at this point because I didn't get the impression that the OP was ready for a discussion about Christoffel symbols.

Why would the Earth accelerate upwards?

Remember, you can only do the inertial reference frame locally in curved spacetime, so it is not the whole Earth accelerating upwards, but just the small patch of the surface of the Earth in the local inertial reference frame, as I said above.

It accelerates upwards because there is a normal force pushing up on it from below which (in the local inertial frame) is not balanced out by other force. Therefore, it accelerates according to Newton's 2nd law.

 

[crossword.bob]

Yes thank you for the reminder, I understand it. I just took out "time" on purpose so that it could be simplified, and that it is harder to visualize time as another dimension. But I can see from what you're saying, time has something to do with the mechanics, and is an important factor. So pardon me.
(On another topic) So this explains time dilation for photons and light itself?

In a loose sense, treating time as a fourth dimension, and disregarding curvature for now (so SR), everything is always moving at a constant speed—the speed of light. If you are at rest in a given frame, then in that frame, 100% of your motion is along the time axis. Now, say you start traveling at a fixed rate along the x-axis (according to the laboratory frame). To do so, while still maintaining the universal speed of light, you have to steal some of your velocity in the time direction to spend it along the x axis—just as if in normal space you take a vector parallel to the y-axis and rotate it in the x-y plane without changing its magnitude. This explains why your clock seems slow to a laboratory observer—you are quite literally not traveling through time as fast! If you had zero mass, you could travel perpendicularly to the time axis at the speed of light; time would be essentially frozen, as for photons.

Now if you can get your head round that, you can see where curvature comes in—while you (and the Earth) are zooming along the time axis in what seems to each of you to be a straight line (i.e. free-fall), you will nevertheless end up meeting, just as two explorers traveling North will eventually cross paths. Hopefully you'll get along.

Edit: or alternatively, so we can learn to pick ourselves up.

 

[stevendaryl]

A. Consider a freely-moving object when there is no gravity. We know such an object moves in a straight line at a constant velocity (relative to any inertial frame). If we draw a graph of distance against time on a flat piece of paper, we get a (red) straight line. Even if the object is at rest in the frame, we get a line parallel to the time axis, not a point.

Yes, that's the key point: every object has velocity along the time axis, so spaceTIME curvature causes even "at rest" objects to veer away from the path they would have without gravity.

If only space is curved, then there would be no gravitational force on objects at rest. But if spacetime is curved, then all objects are affected.

 

[andrewkirk]

kweba I think it would help if you familiarised yourself with the concept of a 'four-velocity'. When you understand that concept you will see that there is no such thing as being 'at rest'. Every object has a nonzero four-velocity. Hence it has a unique geodesic that it follows - if it is free from non-gravitational forces. That geodesic is defined by its position and the direction in spacetime of its four-velocity.

ETA: looks like somebody has just explained this above, with diagrams too. Nice!

 

[DaveC426913]

Now if you can get your head round that, you can see where curvature comes in—while you (and the Earth) are zooming along the time axis in what seems to each of you to be a straight line (i.e. free-fall), you will nevertheless end up meeting, just as two explorers traveling North will eventually cross paths. Hopefully you'll get along.

That is an interesting and very succinct way to explain it.

In nonlinear geometry (positive curvature), lines that start off parallel will eventually cross. So, Earth moving through time and the object moving through time both start off parallel, but will eventually intersect - despite the fact that neither had any initial spatial movement wrt each other.

Dr Greg's diagrams actually demonstrate the same thing, but your explanation is even more elegant. (A few words are worth a thousand pictures?)

Thanks.

 

[TrickyDicky]

A. Consider a freely-moving object when there is no gravity. We know such an object moves in a straight line at a constant velocity (relative to any inertial frame). If we draw a graph of distance against time on a flat piece of paper, we get a (red) straight line. Even if the object is at rest in the frame, we get a line parallel to the time axis, not a point.

B. Now consider the same freely-moving object with no gravity being observed by an accelerating observer. This can be represented by the same flat piece of paper as before with a red straight line on it, but now with curved blue grid lines instead of straight grid lines. If the red line starts off parallel to one of the curved blue gridlines, it won't remain parallel for long. In other words an object released from rest will appear to "fall" relative to the accelerating observer.

C. Finally, consider a freely-moving object falling under Earth's gravity. Now we need to draw our distance-against-time graph on a curved piece of paper. The object will now follow the straightest (red) line possible (a geodesic) on the curved sheet. The blue grid lines representing something at rest relative to the Earth do not follow the straightest routes on the sheet. If the red geodesic line starts off parallel to one of the blue non-geodesic gridlines, it won't remain parallel for long. In other words an object released from rest will fall.

This is a nice way to see it and certainly describes why we fall, but actually if one considers B (that is in flat spacetime), it begs the question why curvature is needed at all for GR and gravitation; B would only need a mechanism of acceleration. So once we assume gravitation comes from spacetime curvature, C is not adding anything to the question why we fall.

 

[Jimmy]

This is a nice way to see it and certainly describes why we fall, but actually if one considers B (that is in flat spacetime), it begs the question why curvature is needed at all for GR and gravitation; B would only need a mechanism of acceleration. So once we assume gravitation comes from spacetime curvature, C is not adding anything to the question why we fall.

How would the object in Example C fall if there was no curvature? I'm not entirely sure I understand the issue.

 

[TrickyDicky]

How would the object in Example C fall if there was no curvature? I'm not entirely sure I understand the issue.

It is probably just nitpicking. I was only pointing out that in B you already have the object "falling" without having to curve the graph as in C, just by introducing acceleration (curvilinear coordinates) making C unnecessary.

 

[Jimmy]

I understand; you feel C is redundant.

However, taken together, B and C seem like a good example of the equivalence principle. And in the context of the original post, I think scenario C is nice to have.

 

[TrickyDicky]

I understand; you feel C is redundant.

However, taken together, B and C seem like a good example of the equivalence principle. And in the context of the original post, I think scenario C is nice to have.

Certainly.

 

[Dale]

This is a nice way to see it and certainly describes why we fall, but actually if one considers B (that is in flat spacetime), it begs the question why curvature is needed at all for GR and gravitation; B would only need a mechanism of acceleration. So once we assume gravitation comes from spacetime curvature, C is not adding anything to the question why we fall.

Spacetime curvature is needed only to explain tidal effects of gravity. You are correct that bulk "falling" has little to do with curvature.

 

[Jimmy]

@Trickydick I'm sorry. I really don't mean to be pedantic. I should be in bed and the "why curvature is needed at all for GR" kind of threw me off.

Spacetime curvature is needed only to explain tidal effects of gravity. You are correct that bulk "falling" has little to do with curvature.

Damn, just when I think something makes sense...

 

[TrickyDicky]

On the other hand it is difficult to represent graphically these things, and unfortunately the time dimension in the graph has to be represented as a spatial dimension and we are actually looking at a 2D space instead of a 2D spacetime, so that in C the curvature of the graph that should represent a spacetime curvature, is in fact a purely spatial one and thus it may lead to some confusion for the not well acquainted.

 

[Jimmy]

I see. I figured it couldn't be that simple. :P

 

[DaveC426913]

On the other hand it is difficult to represent graphically these things, and unfortunately the time dimension in the graph has to be represented as a spatial dimension and we are actually looking at a 2D space instead of a 2D spacetime, so that in C the curvature of the graph that should represent a spacetime curvature, is in fact a purely spatial one and thus it may lead to some confusion for the not well acquainted.

I dunno, I think people are pretty familiar with seeing time plotted on a graph. It's not showing time as a spatial dimension; it's simply representing time as an axis on the graph.

 

[TrickyDicky]

It's not showing time as a spatial dimension; it's simply representing time as an axis on the graph.

I'm making a (maybe too subtle) distinction between what it represents and what the literal picture shows in the context of the nuances of space vs. spacetime.
We all agree that this axis represents the time coordinate, but the graph cannot easily show the difference between a time coordinate and a space coordinate and so it depicts them in the same way, and we see a 2D SPATIAL surface that represents a 2D Lorentzian Spacetime that is basically a 1D space in time.

This might verge on the pedantic, but it could be a source of confusion for some (precisely those that are very familiar with seeing time plotted in graphs but not so much with Lorentzian manifolds and relativity) , and that is why I mention it.

 

[Fredrik]

Why would the Earth accelerate upwards?

What you're asking here is pretty much the same as "Why is GR a good theory?". You're focusing on a specific aspect of it, but I think the only good way to answer your question is to explain why the predictions of GR are accurate, and the only thing that can do that is a better theory of gravity.

Is it true spacetime push as back to Earth?

No, Earth is pushing you away from the part of space you'd have to be into do geodesic (i.e. non-accelerating) motion.

 

[kweba]

kweba I think it would help if you familiarised yourself with the concept of a 'four-velocity'. When you understand that concept you will see that there is no such thing as being 'at rest'. Every object has a nonzero four-velocity. Hence it has a unique geodesic that it follows - if it is free from non-gravitational forces. That geodesic is defined by its position and the direction in spacetime of its four-velocity.

ETA: looks like somebody has just explained this above, with diagrams too. Nice!

Yes, thank you. I admit I am not familiar with the concept (I never heard of it.) I guess this is what I'm missing all along. Thanks, I will read/study about it.

But I never realized, until now, that we are in a constant motion (velocity) through spacetime, that even though we may be at rest in space, we are still in motion through time.

It said that our four-velocity, with everything in the universe, is at the constant speed of light, but it just goes to the time direction. Is this true?

Yes I tried studying the graphs, but pardon me as I'm having a hard time getting it.

 

[Fredrik]

Yes, thank you. I admit I am not familiar with the concept (I never heard of it.) I guess this is what I'm missing all along. Thanks, I will read/study about it.

But I never realized, until now, that we are in a constant motion (velocity) through spacetime, that even though we may be at rest in space, we are still in motion through time.

It said that our four-velocity, with everything in the universe, is at the constant speed of light, but it just goes to the time direction. Is this true?

It's useful to understand four-velocity, but I don't think it will help you answer the question in the thread title. What you need to understand is that SR and GR both involve a mathematical thing called "spacetime", and that motion of particles is represented by curves in spacetime, called "world lines". There's a class of curves that really distinguish themselves from all the rest, in a way that doesn't depend on a choice of coordinate system. These curves are called geodesics.

Since they are the only world lines that are "special" in a coordinate-independent way, it's natural to define the proper acceleration of a world line in a way that ensures that the proper acceleration of a world line, at an event E on it, is a measure of much the curve "curves away" from a geodesic through E that's tangent to the world line. The force acting on the object can then be defined by F=ma. To find the geodesics near a spherical distribution of mass (like a planet), we must solve Einstein's equation for that distribution. When we do, we find that the curve representing the motion of an object on the surface isn't a geodesic, so the a (acceleration) of the object is non-zero. Since F=ma by definition, this means that there's a force acting on it. Since the surface of the Earth is preventing the object from doing geodesic motion, we say that the surface of the Earth is pushing it with force ma.

(What I wrote as F=ma is actually a four-vector equation. I'm trying to keep things simple, so I won't say anything more about that).

The four-velocity of a world line is defined as the normalized tangent vector of the world line. This makes the statement that everything moves through spacetime at speed c trivial and not interesting in my opinion.

 

[A.T.]

Yes I tried studying the graphs, but pardon me as I'm having a hard time getting it.

Falling from rest:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

Vertical throw upwards (the second half is the same as falling from rest):
http://www.relativitet.se/spacetime1.html