Why do apples fall rather than lift off the ground?

Recommended Posts

Well GR 's equations show(I have to take this on trust as I have not learned them personally) that sources of mass and energy curve neighbouring spacetime with the result that bodies move along  the geodesics  and that we see as them "falling" towards that source ,when account is taken of their existing momentum.

Why do these bodies move "down" rather than "up"?

Is there something in the equations  that only allows them to ,as it were be attracted rather than repelled by the sources of mass-energy?

Share on other sites

GR reduces to Newtonian gravity in the weak field limit, and that’s attractive, so...yes.

Share on other sites

I've been thinking for like half a minute whether there could be any other reason than the Newtonian limit, but I can't find any other one.

Share on other sites

I'm sure Markus or Mordred will give you a much more detailed answer when they log on tonight..

All I can offer is that a geodesic is an orbit, a 'fall' is essentially an arc of that orbit, and the speed and direction of a test mass are defined by the energy and momentum of that test mass.
IOW, an orbit could be circular,  or straight through the gravitating body ( assuming point sources ),  'up' the other side, and back down again; its initial conditions determine direction along the orbit, or geodesic.

Share on other sites

Through spacetime, any geodesic traces a (locally) straight line, not curved at all.  So a bullet going from me to a target follows a straight line, not a parabola.  Similarly, the ball tossed in what appears to be a high arc to the basket also traces a straight line through bent (non-Euclidean) spacetime,.

The apple doesn't fall up because spacetime isn't bent in a way that would allow that to be a straight line.

Share on other sites
5 hours ago, geordief said:

Why do these bodies move "down" rather than "up"?

They fall “down” due to a principle called the principle of extremal ageing. This principle, in simple terms, implies that geodesics in spacetime tend to be the longest (in geometric terms) they can be, given all initial and boundary conditions. Since in an environment such as planet Earth the length of a free-fall geodesic is generally dominated by the time-term within the metric, this implies that such geodesics will tend to be oriented towards regions with higher time dilation, relative to some reference point far away. In other words - closer to the central body. This is why the apple falls down, rather than up - because this is what maximises the geometric length of its world line, given its initial and boundary conditions.

It’s important to bear in mind the bit I have highlighted in the end, because initial and boundary conditions do of course play a role in this - for example, if you were to throw the apple upwards with enough acceleration, it might escape to infinity, rather than fall down, because given those initial conditions, that’s the longest possible world line for it. It’s also important to remember that our intuition of what constitutes “long” is once again considerably different from the maths - the purely spatial distance from tree to ground appears very short to us, whereas in fact its geometric length in spacetime is considerable; that’s because in the metric, the time part carries a factor of $$c^2$$, which often makes it greater than the spatial part by some orders of magnitude.

Mathematically, geodesics are solutions to a set of partial differential equations, hence initial/boundary conditions determine the form of the solution just as much as the equation itself. The equation is, in fact, just the principle of extremal ageing written in mathematical form.

Share on other sites

Markus beat me to it with his excellent explanation.

Here is a pdf on the principle of extrenum aging.

Share on other sites
Posted (edited)
10 hours ago, Markus Hanke said:

They fall “down” due to a principle called the principle of extremal ageing. This principle, in simple terms, implies that geodesics in spacetime tend to be the longest (in geometric terms) they can be, given all initial and boundary conditions. Since in an environment such as planet Earth the length of a free-fall geodesic is generally dominated by the time-term within the metric, this implies that such geodesics will tend to be oriented towards regions with higher time dilation, relative to some reference point far away. In other words - closer to the central body. This is why the apple falls down, rather than up - because this is what maximises the geometric length of its world line, given its initial and boundary conditions.

In my first post on this site, I was puzzled as to why an apple seeking to accumulate the greatest proper time on its wristwatch, would head ''down'' to where time is slower and hence slowing its accumulating of proper time. In other words, the apple is stopping its ageing by going ''down''.

Compounding my puzzlement, is that time dilation in the schwarzschild coordinates is only between stationary observers, that is, a stationary far away observer and a stationary ''shell'' observer near a black hole.

So, for the falling apple there is no time dilation to travel down to, so to speak.

If there was a slowing of time for the apple it would never reach the event horizon. help someone.

I'm muddled in my thinking and know it's down to my incorrect understanding of things.🙂

Edited by crowman

Share on other sites
9 hours ago, Markus Hanke said:

They fall “down” due to a principle called the principle of extremal ageing. [...]

Ok. Just one question, though. Can't that be obtained also from the fact that the Newtonian approximation always ties you to some g00 that to first order must have a Newtonian interpretation and thus the maximum aging (or maximum proper time) principle is already implied by it? IOW, isn't it a matter of what axiomatic approach you take?

As not all fields are amenable to weak field approx., what you say does seem more general and preferable from an axiomatic POV...

On the other hand, you can always do a Taylor series expansion and the first order is one plus the Newton potential, according to the historical approach.

Share on other sites
Posted (edited)

The Principle of maximal aging as it is also called is reducible to Newtonian mechanics including the principle of least action. So both answers are valid.

Edit keep in mind under GR freefalling objects ie constant velocity have no force acting upon it. Force of gravity being described by spacetime curvature. Ie an accelerating worldwide.

Edited by Mordred

Share on other sites
On 6/2/2020 at 5:42 PM, geordief said:

Is there something in the equations  that only allows them to ,as it were be attracted rather than repelled by the sources of mass-energy?

Quote

Under general relativity, gravity is the result of following spatial geometry (change in the normal shape of space) caused by local mass-energy. This theory holds that it is the altered shape of space, deformed by massive objects, that causes gravity, which is actually a property of deformed space rather than being a true force. Although the equations cannot normally produce a "negative geometry", it is possible to do so by using "negative mass". The same equations do not, of themselves, rule out the existence of negative mass.

[...]

Bondi pointed out that a negative mass will fall toward (and not away from) "normal" matter, since although the gravitational force is repulsive, the negative mass (according to Newton's law, F=ma) responds by accelerating in the opposite of the direction of the force. Normal mass, on the other hand, will fall away from the negative matter. [...]

The Standard Model of particle physics, which describes all currently known forms of matter, does not include negative mass.

Also, you're getting a lot of answers to "why" things fall toward each other. If people didn't have answers, you'd probably see something like "Science doesn't deal with 'why', it deals with 'what'", and that applies here too. The answers aren't the root cause of attraction or anything like that, they're just other measurements that correspond. Masses correspond with a certain geometry of spacetime, geodesics of that geometry have certain configurations. If 'what' makes sense enough, you tend to stop thinking about 'why' (eg. one wouldn't wonder "why can't a circle curve outward instead of in on itself?" if one knows what a circle is).

Share on other sites
Posted (edited)

As to geodesic paths three good guidelines to follow.

1) The speed limit of all possible exchange of c. (This also applies to possible worldlines.

2)  A field in GR amounts to a set of potential values at each event.

3) greater potential difference in the field potentials result in greater energy/momentum. ( A curved worldine is a worldine with acceleration) (has rapidity) which can be screw symmetric) a freefall worldline has constant velocity)(symmetric under Lorentz transforms)

Most important mass is simply the resistance to inertia change or acceleration while energy is the ability to perform work. The classical definitions do not change.

Of fundamental importance time is described by the interval between two events. Ie time for a signal to transverse between those two events.

Edited by Mordred

Share on other sites
Quote

Why do apples fall rather than lift off the ground?

Leonard Susskind would say that there are existing parallel Universes in which everything is reversed and "apples lift off the ground" (but that would disallow creation of "apples" in the first place, so we have contradiction/paradox)..

Share on other sites
5 hours ago, Sensei said:

Leonard Susskind would say that there are existing parallel Universes in which everything is reversed and "apples lift off the ground" (but that would disallow creation of "apples" in the first place, so we have contradiction/paradox)..

IIRC I have been told on this same Forum that under time reversal Gravity is still attractive. Which makes some sense I have to admit.

Share on other sites
1 hour ago, michel123456 said:

IIRC I have been told on this same Forum that under time reversal Gravity is still attractive. Which makes some sense I have to admit.

Yes, that's right. Time reversal in the equations of motion for a falling object just make it look like an escaping object by reversing the initial conditions for velocity. While the force term remains the same.

6 hours ago, Sensei said:

Leonard Susskind would say that there are existing parallel Universes in which everything is reversed

Everything? I also think an invitation proffered to Leonard Susskind to the forum to give him a chance to actually say what he would actually say is in order.

Share on other sites
On 6/4/2020 at 5:48 AM, Mordred said:

(...)

Edit keep in mind under GR freefalling objects ie constant velocity have no force acting upon it. Force of gravity being described by spacetime curvature. Ie an accelerating worldwide.

constant velocity?

Share on other sites

Yes constant velocity for flat spacetimes. Curved spacetime is a curved geodesic and includes acceleration.

Share on other sites
Posted (edited)
On 6/5/2020 at 2:50 PM, Mordred said:

Yes constant velocity for flat spacetimes. Curved spacetime is a curved geodesic and includes acceleration.

Edited by michel123456

Share on other sites
Posted (edited)

Newton's first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.

This describes a freefall condition. It also describes a state where you have either constant velocity or at rest.

Now the Lorenz  transforms describes the relativistic affects under a constant velocity.

See here this link saves me typing the transformation rules.

Now any change in velocity or direction involves acceleration. In the Lorentz transforms this is described as rapidity. It will involve a scew symmetry of the Lorentz transforms such as a rotation or boost.

Proper acceleration (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity.

A straight line geodesic doesn't involve acceleration. Objects that experience no force including psuedoforces move at constant velocity.

This isn't a curved spacetime.

A curved spacetime describes the particle path when you involve acceleration. Hence curved spacetimes are assymetric. A flat spacetime under constant velocity is symmetric under a 180 degree rotation of the momentum vector.

Example a car moves 180 km/hour in the x axis direction. If the car maintains the same velocity in the minus x direction. You have a symmetric relation where the only difference is the change in the plus or minus x direction.

When you involve acceleration you now have a curved geodesic or worldwide. (Curved spacetime ). A curved path must involve acceleration as you have direction changes at the minimal.

Edited by Mordred

Share on other sites
4 hours ago, Mordred said:

Newton's first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.

This describes a freefall condition. It also describes a state where you have either constant velocity or at rest.

I get it wrong from the start. To me "free fall" is accelerated motion. Not constant motion.

I can understand that GR considers free fall as constant motion under the prism of a geodesic in curved spacetime. But what is constant motion then? And what is standing still?

Share on other sites
Posted (edited)
On 6/8/2020 at 3:52 AM, Mordred said:

When you involve acceleration you now have a curved geodesic

While I understand what you are trying to say, may I just point out here that geodesics are never ‘curved’ in the Euclidean sense, since by definition they are world lines that parallel-transport their own tangent vectors at every point, so they are always locally ‘straight’. They also never involve acceleration, since they must be solutions to the differential equation

$x{^\mu}(\tau){_{||\tau \tau}}=0$

I know I’m just nitpicking, but this bit is crucially important in GR

Edited by Markus Hanke

Share on other sites
Posted (edited)

Good point just a side note I've often wondered if the Frenet Serett equations used in R^3 would apply to R^4. However that's just a side note curiosity lol.

Edited by Mordred

Share on other sites
On 6/9/2020 at 5:54 AM, Mordred said:

Good point just a side note I've often wondered if the Frenet Serett equations used in R^3 would apply to R^4. However that's just a side note curiosity lol.

I know that these relations can be generalised to any number of dimensions on Euclidean manifolds; I also know that they can be generalised to Minkowski spacetime. However, whether one can generalise them to arbitrary pseudo-Riemannian manifolds is a question I don’t know the answer to (probably best posed to a mathematician). I suspect that, if it is possible at all, you’d end up with something very convoluted and awkward, certainly something much more difficult to work with than the comparatively simple geodesic equations.

Note that the Frenet-Serette equations apply to any smooth, differentiable and continuous curve, so for our purposes you’d need an additional constraint to pick out geodesics on your manifold.

Share on other sites

Because the ground got there first, by falling.
If apples "fell up" then the ground would too.

It would be on the ceiling, but the apples would still land on it if you let go of them.

Share on other sites
8 hours ago, Markus Hanke said:

I know that these relations can be generalised to any number of dimensions on Euclidean manifolds; I also know that they can be generalised to Minkowski spacetime. However, whether one can generalise them to arbitrary pseudo-Riemannian manifolds is a question I don’t know the answer to (probably best posed to a mathematician). I suspect that, if it is possible at all, you’d end up with something very convoluted and awkward, certainly something much more difficult to work with than the comparatively simple geodesic equations.

Note that the Frenet-Serette equations apply to any smooth, differentiable and continuous curve, so for our purposes you’d need an additional constraint to pick out geodesics on your manifold.

Agreed most likely adds additional complexity. Might dig further into it though just to satisfy my curiosity.

Create an account

Register a new account