Understanding General Relativity

A Troubled Man

Hi all,

I'm having a discussion with someone who believes they understand GR, here is their understanding of how gravity works...


"The time and space at your feet are different from the time and space at your head.

This has been experimentally verified, especially in terms of the time component. We have the technology to measure time accurately enough to actually make these kinds of experiments, and they have been made. A clock placed at your feet will run slower than a clock sitting on your shoulder.

It is the dilation in spacetime (both space and time) between your head and your feet that cause your body to accelerate downward toward the massive earth.

And because this is indeed the cause of gravity on the Earth, this means that you body is actually exerting a downward force onto the Earth's surface."

BenG549

Distance form a gravitational source will have an effect on time, so your head ages at a different rate to your feet!

Hmm... The gravitational force can be thought more of as the bending of spacetime by an object. If you imagine placing a spherical object on a simply supported large piece of paper it would create a similar "distortion" and things would effectively "fall into" the hole towards the object... its a similar concept... if you travelled fast enough around the "side" of this slope, you would effectively orbit the object. It's hard to visualise in 4 dimensional spacetime though.

We do exert a gravitational force on the earth, if you look up the equation you can establish how much the earth is "pulled" up towards you if you were to jump... it's obviously negligible, but it does happen.

DaleSpam

This is essentially correct. Was there something in particular that bothers you about this description?

A Troubled Man

How is it that gravity is the result of the difference in time dilation between ones' hand and head?

How can a body in free fall be exerting a force? Isn't the surface of the earth an accelerated frame, hence it is the surface accelerating upwards towards the body in free fall?

PAllen

Forces balance. Earth's surface exerts force on you, holding you stationary. You exert force on earth, resisting being prevented from free falling. The two forces are equal.

Ken G

That's an important part that your friend did not explain! It comes down to what constitutes an "inertial path", i.e., the path in spacetime taken by an object with some initial motion that has no forces on it (gravity is not a force here). The key point about inertial motion is that the motion in the "spatial direction" is perpendicular to the motion in the "time direction" (you move in both space and time, so both count in inertial motion). That's the part that wasn't explained to you-- if time is going faster at your head than at your feet, it forces your motion in x,t space (x is the vertical coordinate of your height above the ground) to curve (so you change x coordinate, not just t coordinate like you might normally imagine for an object that starts out stationary and has no forces on it). That's because the direction that is "perpendicular to time" is curving downward, if time is going faster (in a relative way) at your head than at your feet. This is also called "curvature of spacetime", because one can give a geometric description of it. There are actually many different ways to think about what is happening there, they are all just "stories" being told from the perspective of various different choices of coordinate system (so I would object to your friend's language that this is what gravity is "really doing", if you don't like that picture there are many others that sound completely different on the surface).Your friend is not talking about a body in free fall, but rather the situation of when you are standing on the ground. The surface of the Earth is indeed an accelerated frame, and so is the person standing on it (they can feel the force on their feet). That force is needed to make you follow a noninertial path-- the inertial path is the one where you are in free fall, and if you are in free fall, you exert no forces on the Earth at all (though you do slightly curve the spacetime the Earth is moving through).

DaleSpam

Here is a rough analogy. Suppose that you have an axle with two wheels rolling across a surface. If both wheels are rotating at the same speed then you are going straight, but if one wheel is rotating slower than the other then you are going to be turning.

There is a precise geometric sense in which we are "travelling" through time (or rather spacetime), and the fact that the clock at your feet is "rotating" slower means that your path through spacetime is "turning". The only way to stop "turning" is to go into free-fall where you wouldn't detect any difference in time between your head and your feet, but the force of the ground on your feet makes you keep "turning".

A body in free-fall is not exerting a force. However, a body on the surface of the earth is not in free-fall, it is being accelerated upwards. By Newton's 3rd law it doesn't matter which object you say is exerting the force. The upwards force on your feet is equal and opposite the downwards force on the earth.

Bill_K

The gradient of time dilation is the gradient of g₀₀. But for a static field in linearized GR, g₀₀ = 1 - 2Φ, where Φ is the Newtonian potential, and so the gradient of g₀₀ is also the Newtonian gravitational force.

A.T.

See the first picture here:
http://www.physics.ucla.edu/demoweb/...spacetime.html

A Troubled Man

He said that we exert a force on the Earth while in free fall.

A Troubled Man

Would a test particle follow the same path considering it doesn't have that same difference in time dilation?

That is exactly what I was objecting.

That's what I was trying to explain but he told me I was out to lunch claiming the surface couldn't possibly be an accelerated frame because that would mean the surface of the earth would be expanding outwards,the earth would grow in size continuously.

A Troubled Man

Again, as asked above, would a test particle follow the same path? As well, considering that the difference in time dilation between one's head and feet is infinitesimally small, how can it account for such a major role in the path followed?

But, in relativity, it is the surface that is accelerated upwards and not the object exerting a force on the surface.

Bill_K

Because, as I pointed out in #8 above, it is not the difference in time dilation, it is the gradient of the time dilation that represents the Newtonian gravitational force.

A.T.

It is a common misconception, that acceleration towards/away from a point implies a change in distance to that point. It simply doesn't. Not even in flat space-time where you can have circular motion with a centripetal acceleration towards a center, without ever coming closer to the center. In curved-spacetime the opposite case is also possible: The Earth surface can accelerate away from the the Earth's center, without getting further away from it.

A.T.

The gradient in the time-dilatation dominates only for objects that move much slower than light through space. Such objects move mostly through time, so the distortion of the time-distances has the most effect.

If the object is in freefall it doesn't exert any forces on the surface in GR. If the object rests on the surface, they exert equal but opposite contact forces on each other

Devils

That's always confused me as well.

So I am sitting here perfectly still but accelerating towards the Earth at 9.8m/s^2. How does classical mechanics or GR explain that? Newton certainly knew about it.

WannabeNewton

Well it is quite easy to explain with a classical analysis because the reaction force (normal force) from the ground on you tends to cancel out all normal components of force acting on you if you treat your motion as being at rest. You don't even need to invoke GR for that.