Something recently reminded me of a fun question that I loved to taunt folks with back in grad school. It wasn’t an apple falling on my head, but close enough. It’s fun because the answer is simple (and there aren’t many choices), but the justifications of the answers often border on the epic.
To make this more interesting, the first person with the correct answer AND and an explanation that I like will get a free Tech-Recipes “Computers Fear Me” t-shirt. If we all have fun with it, there may be more quizes in the future. So here goes:
According to Einstein’s equivalence principle, the effects of a gravitational field are indistinguishable from that of linear acceleration. So, in what direction are we accelerating as a result of Earth’s gravity?
25 comments ↓
It’s been awhile since I had physics so I appologize if this answer is completely wrong; but maybe you’ll get a decent laugh… who knows?
We are accelerating in a tangential direction on a plane at 90 degrees with the rotational axis of the Earth that just happens to be following the Revolutions of the Earth around the Sun. Basically, we’re like a giant Spirograph. We aren’t accelerating towards the center of the Earth as long as we’re on it’s surface due to the Normal Force of the Earth’s Surface and even that acceleration would only be Relative to our location on the Earth. If you look at it relative to the location of the Sun, it’s more of the Spirograph movement that I mentioned earlier.
Maybe I’m completely off on a tangent myself but I must return to work now.
Oh, man! I absolutely love the idea of the giant Spirograph! I’m also picturing the moon’s spirograph, too. Very cool.
So (besides the awesome tangent on toys of our youth) you’re saying we aren’t accelerating right now because of the Earth’s surface holding us in place? Not the answer I’m going for, but you are raising good thoughts about different frames of reference…
the answer is simple: we aren’t accelerating. We *are* in motion though, and that motion is effected by gravitational forces. Nice trick question.
Earth’s gravitation field accelerates us straight down, toward the center of the mass of the Earth. The only reason this seems tricky is because you mentioned Einstein (and that alone will make people over-think) and because people like to mix in other vectors. A lot of people have read about how an orbiting object has a tangential vector and “fallsâ€? into it’s orbital path. Or they read about how just standing on the surface of the Earth, we go spinning and flying around in all kinds of directions, taken from a wider perspective. All of those things are true, but you asked about this one vector alone. The gravitational force is very simple and straight down.
Another point - just because we are under the influence of the gravitational vector, doesn’t mean we move in response to it. The surface of the Earth prevents our feet from passing through it and so we stay on the surface. If the ground beneath our feet disappeared, we would fall toward the center of the Earth under thge influence of the gravitational acceleration vector.
So we have another “not accelerating” and a new “down” answer, neither of which are correct. No hints yet! (although two of the three obvious answers have been shot down).
A late bit of fine print: anyone that has played this one with me, and you know who you are, are ineligible. Many a bottle of wine have given their all in the pursuit of this simple question. Okay, a hint: if you drink wine while playing through this quandary, the corks are useful for problem solving props.
According to Einstein, there is no such thing as gravity, there is only curved spacetime. Under no acceleration, objects move in straight lines, but because spacetime is curved near massive bodies the paths do not appear straight. On the surface of the earth, we are being accelerated *upwards*. This acceleration causes us to move on a curved path through spacetime, but the curvature of our path precisely matches the curvature of spacetime so that the resulting path appears to be stationary in space (relative to the earth). The physical sensation of standing on the surface of the earth is indistinguishable from the physical sensation of being accelerated upwards (i.e. towards the tops of our heads) at 9.8 m/s^s.
Ron (above) is correct.
An attempt at more lay terms:
If you stand in a room in space and the room is pulled on by an outside being who tied a rope to your “ceiling”, then you accelerate towards the bottom of the room, and (for all you know) you are experiencing a “downward gravitational force” — similar to that on earth. Relative to “space” (outside the room), however, you are actually accelerating “upwards” (along with the rest of the room). I say “upwards” in quotes because it is “up” relative to the floor — which is analogous to the surface of the Earth.
Awwwwhhhhh, shucks. I wanted to play!
Except that it is not gravity that is providing the upward acceleration that you are talking about. It is the surface of the Earth (or the floor, or whatever you are standing on) which is pushing up through your feet to make you stand more-or-less still (actually, even when you are stationary, you are accelerating down towards the center of the Earth. That’s because the Earth is spinning, and so gravity actually is accelerating you at a rate equal to the centripetal force).
So from a General Rel point of view, gravity is not accelerating you in any direction, but the surface of the Earth is accelerating upwards at around 9.8m/s/s minus the centripetal acceleration of the rotation of the Earth.
Here is my take:
You state that acceleration and the gravitational fields are equivalent. That means, in a sealed box with no windows, you could not tell the difference between the box being accelerated by an outside force in space, and the box just sitting there on the ground of Earth.
However, just because they are equivalent, it does not mean you can simply swap one for the other. When we stand on the Earth, we *know* we are stuck there by gravity. We *know* we are not in an accelerating box. So, it makes no sense to say we *are* accelerating when we know we are not.
What we can say, is that the forces we experience are equivalent to being accelerated away from the Earth’s centre at 9.81m/S2. But as for actually accelerating, the answer is simple: we are not.
– Jason
Actually, Matt
2 Things:
1. What you said about centripetal force can’t be right: Earth is spinning, but gravity does not accelerate you at a rate equal to the centripetal force. If that was true, you’d fly off the earth because of its spin. Recall that in a rotational system the vector of velocity is tangent to the circular path of the object. So this is definitely not the case. Gravity just pulls you “down” towards Earth in Earth’s frame of reference.
2. The answer to this question is still “upwards”:
True, *relative to the earth*, gravity is accelerating you towards Earth. This is not what the question asked. In general (hence the name “general relativity”), you are still being accelerated “upwards”, but you only appear stationary on the surface of the earth because the earth is your frame of reference. You wouldn’t assume it as an Earthling, but standing on Earth is just the same as standing on a virtually “massless” planet which is pulled up from under you by some other force.
If you think about it again in terms of the room in space analogy I posted earlier, the person in the room does not know that it is the room that is being pulled by a rope. As far as he is concerned, there is a gravitational force coming from the floor, because he accelerates, in the frame of reference of the room (earth), towards the floor (surface). In the frame of reference of the “person” pulling the rope — or any other outside observer at that (which consitutes a more “universal” frame of reference) — the man in the room is still accelerating upwards, in the direction of the pull of the rope.
Oh, yeah, we have a winner! And it’s not Davak.
Ron correctly identified that up is the direction in which we’re accelerating as a result of Earth’s gravity and gets the shirt! Ray adds a nice explanation of why up is right.
I’ll add my musings about it, too. Are we really accelerating standing still on the Earth’s surface? Yes! If you make a simple accelerometer with a spring and a mass at the bottom, in Ray’s room that is resting on the Earth’s surface, your accelerometer would be deflected downward.
If the room is moved into space far away from any gravitational or other forces, the accelerometer will not be deflected at all. As ray points out, with a rope pulling the room “upward,” the acceleration will be the same as experienced on Earth, accelerometer deflected downward.
Another interesting case is if you let the room fall freely toward a hypothetical Earth with no atmosphere, the accelerometer will show no deflection - no acceleration (despite accelerating toward the Earth’s surface at 9.8 m/s/s). This is the only reference system that makes sense to use for our discussion, a nice inertial reference system.
The room sitting on the Earth constitutes a non-intertial reference frame in which the force you feel on your feet is the ficticious force of gravity because of the (upward) acceleration acting on the reference frame (room).
Ahh, now the comments become interesting. Jason picks the non-interial reference frame of the Earth’s surface to make his decision that we are not accelerating. Yeah, like someone can bend statistics to justify something false, you could argue that I’m picking my reference system to justify my up answer. But is it wise to pick a reference system in which Newton’s first two laws don’t apply?
One other common sense approach to this: in which direction must an elevator accelerate to add to the feeling you experience by Earth’s gravity?
So I’m stood on the ground here in the UK. Someone else is stood on the ground in Australia. We are both accelerating away from each other (away from the centre of the same Earth) and yet we remain the same distance apart from each other?
I always thought that acceleration was the rate of change in the velocity of an object. If what you say is true, and two objects can accelerate away from a single point, in opposite directions, and never get further apart, then my understanding of the world is way out. I may just fall off the Earth if I’m not careful
Could you explain a little more about what you mean when you say the reference frame I picked (The Earth) is one in which Newtons first two laws do not apply?
– Jason
Sorry to be hogging the replies, but I understand what you are arguing now.
If I stand on the Earth, then I feel a force upward through my feet - the ground pushes up against me.
If we take the assumption that acceleration and gravity (or is it gravitation?) are equivalent, then we can swap the gravitational field for acceleration. So instead of gravity exactly balancing out the force I feel from the Earth through my feet, we can say that there is actually no gravity at all, and instead I am accelerating.
So the force is pushing my feet, so it must be accelerating me upward, toward my head.
If I am accelerating, then I am also moving, since my velocity is increasing my 9.8m/S/S (it is how we define acceleration). I must be moving upward, head first.
Since my feet remain on the ground, the whole Earth must be moving with me - it is accelerating toward my head. Since the gravity between myself and the Earth is not affecting the Earth’s own position in the Universe, i.e. its position in a bigger reference frame, then everything else must be accelerating along with it.
So ultimately, I am at the centre of the universe, which is constantly accelerating in its entirety in the direction from my feet to my head.
As I rotate with the Earth, my orientation with respect to the Universe changes by 360 degrees every 24 hours. So too must the acceleration of the universe. So, just by standing on the Earth in one place, I am shaking the whole universe in a cyclic fashion as it tries to keep up with me.
Thank you for reminding me how important and unique I am, and sorry to all those life-forms across the universe that must be getting very dizzy with my juggling antics
– Jason
Couldn’t you just say that curved space affects acceleration, and therefore gravity does cause acceleration? I understand the technical, definitions based argument but I fail to see why you couldn’t just redefine the curvature of space to cause an acceleration and retain the exact same results for the physical model. Do we lose any phenomenon by making this change to definitions?
Jason, you’re totally bogarting the comments, but we’ll forgive you because your comments are great! However, you’re dead wrong about one thing. *I’m* clearly the center of the universe.
Clarification on the reference frames is a good thing since the references systems make all the difference in this problem. It’s intuitive to pick the Earth’s surface as your reference frame, but it is a poor choice when working with vertical problems. Because Earth’s gravitation field at its surface is perpendicular, the vertical coordinate system there is non-inertial. Any coordinate system in the horizontal plane (parallel to the Earth’s surface) is an inertial reference frame, though.. you aren’t pulled sideways as a result of Earth’s gravity.
So let’s play with horizontal a bit. Let’s say you are sitting in a parked car on the Earth’s surface. The driver floors it and the car accelerates forward. The comparable reference frame for the passenger is the car and from that reference frame, it feels like you are being accelerated backwards, pulled into your seat by some force. This is an example of a ficticious force (a force resulting from the acceleration of the reference frame). The proper reference frame is the Earth (in this example) because it is the only inertial reference frame. Earth perpendicular is not a good reference frame because it is actually accelerating upwards.
I was writing this in response to Jason’s comments, but does it help answer Bob’s, too? If not, I may need a physicist assist.
Am I wrong that the issue here is that the force provided by gravity is not being considered an acceleration, but rather an “object in motion staying in motion” which happens to be in curved space time?
My question is if this model loses any power if you just redefine general relativity to say that “curved spacetime is an indicator of where accelerations will take place”.
Well, all I know is that there should be a bunch of t-shirts given out for a thread this awesome. If I were not at work, I would be bouncing the thread to the front page of techrx too.
Great discussion guys. I’ll keep waiting for the imaginary time discussion next…
[…] My friends over at Tech Recipes have announced a contest to win a free Tech Recipes T-shirt. All you have to do is be the first person to correctly answer some simple question…. about Einstein’s equivalence principle…. […]
Hmmmm…I never knew that about gravity and stuff. Nice to learn something everyday. Yet, all I care about is that when I throw a ball into the air, it is gonna come down, and not fly infinitely up into space.
Jason,
I think once you understand the importance of an intertial reference frame, the answer may not seem so strange. And you don’t need to understand GR to see why the surface of the Earth is a bad reference frame for doing physics.
Consider that the laws of physics should be the same everywhere. Then fill your sink full of water and pull the plug, and watch the water develop into a spinning tornado-like shape as it drains out. Of course if you take the sink then to the other side of the world, the water spins in the opposite direction. So the laws of physics seem to change depending on your location. That’s because the surface of the Earth is not an intertial reference frame (even pre-relativity, the fact that the Earth is spinning means it is not an intertial reference frame).
So in an interial reference frame on the surface of the Earth, the Earth is accelerating upwards at 9.8m/s/s, and in a reference from on the opposite side of the Earth you get the same thing. Which seems to indicate that the Earth should explode, since the two sides are accelerating away from each-other. This is where general relativity and the curvature of space-time cut in. When you take the intertial reference frame at one point (for example, wherever you are standing now), then connect that to the point slightly beneath you, and keep going and connecting points all the way through the Earth until you reach the other side, what you will construct is a 5-dimensional Minkowski space that is curved in such a way that both sides can accelerate away from each other, yet not move. (And don’t even try to picture the curvature in your head - Minkowski space doesn’t follow the same rules as regular Cartesian space so its a little difficult to picture. It takes 8 dimensions to fit GR into regular Cartesian space, and 8 dimensions are pretty much impossible to imagine).
Roy F:
It may seem unintuitive, but any object spinning around a point is constantly accelerating towards the center of rotation. That is called centripetal (towards the center) acceleration. Its easy to see that the acceleration is towards the center if you look at a piece of string with some weight on the end, and spin it around in a circle. You can see that the only possible acceleration that the string can provide to the weight is towards the center. So, as you said, an object on the surface of the Earth would move fly off into space if no forces acted upon it. Gravity therefore provides the centripetal acceleration that keeps objects from flying off into space.
Thanks to everyone for contributing their answers and explanations. My first Quandary was a bigger success than I expected. I put three T-shirts in the mail today destined to Ron Garret for getting the correct answer first, to Roy F. for his intuitive explanation of the answer, and to Cameron for his spirograph comment.
Check out the next Q’s Quandaries!
Thanks a lot you guys. This really helped me understand the concepts of inertial frame and why and how we choose one. I had been thinking about this earlier but all people would tell me is to read GR properly and i didn’t have time and i suppressed my curiosity. Thanks again for this disscusion.
I say upward - away from the earth. That’s it. Plain and simple.
I’m glad I didn’t see this image of the Earth’s uneven gravitational field before posting this quandary. Their scale is way out of proportion so that the finer details stand out, but it’s still cool.
Kushagra, I’m glad that this helped you! I intended it to be fun as it’s something I’ve enjoyed contemplating for years. I didn’t expect anyone to find it and find it helpful! Thanks for dropping by and sharing that.
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