Difference between Newton & GR

[John15]

Can anyone please explain the difference between Newtonian and GR when used to describe gravity especially with reference to gravitational attraction inside an event horizon. Or how do we get over the problem of an object in freefall exceeding the speed of light as it approaches the singularity.

 

[atyy]

Gravity in Newton is described only by one number at each point in space, whereas in GR it is described by 10 numbers at each point in spacetime.

In GR, relative velocities are only defined between objects whose worldlines intersect in spacetime. Let's call that the local velocity. The local velocity of a massive object in freefall never exceeds the speed of light, even as it approaches the singularity.

What goes faster than light in GR? It depends on the definition of "what" and "goes", but there is a sense in which space itself falls faster than light into a black hole.

 

[John15]

f=ma. If force (gravity at singularity) = infinite then acceleration must = infinity. Why does this work outside an Event Horizon and not inside.
If an object hits the EH ,heading directly for the center, at c then why does it not continue accelerating as gravity increases.
It seems to me there are only 3 possibilities
1 It stops accelerating
2 It carries on accelerating
3 It stops at the event horizon.

1 and 2 appear to break the rule that the laws of physics remain the same for all.

 

[PeterDonis]

2 It carries on accelerating

This is the correct answer.

1 and 2 appear to break the rule that the laws of physics remain the same for all.

No, they just tell you that "f = ma" is not the correct law of physics. The correct law is the Einstein Field Equation, and for this particular case the Schwarzschild solution to that equation. The paper atyy linked to gives a good intuitive way of picturing what the Schwarzschild solution describes.

As far as the difference between the Newtonian and GR description of gravity, the big difference is that in GR the definition of "force" is different. In GR, a "force" is something you can actually feel; more precisely, it's something that causes an accelerometer moving along with you to register a non-zero reading. Gravity does not do this; an object moving solely under the influence of gravity is in free fall and an accelerometer moving with it reads zero. So in GR gravity is not viewed as a force; it's viewed as a sign that spacetime is curved.

The curvature of spacetime also means that, as atyy said, you can't really assign any meaning to the "relative velocity" of objects that are spatially separated. So even though it looks like a freely falling object is moving "faster than light" once it goes inside a black hole's horizon, that isn't really a meaningful statement. The rule in SR that "nothing goes faster than light" is generalized in GR to "nothing moves outside the light cones"; the light cones are surfaces in spacetime formed by the worldlines of light rays radiating from a particular event. This formulation also works in SR, where spacetime is flat (this should be obvious if you think about it), but it generalizes easily to curved spacetime, since we can just plot the paths of light rays, even if they look curved instead of straight, and use them to define the boundaries of where objects can travel. When we do this for objects freely falling into a black hole, we find that they are still moving inside the light cones; it's just that the light cones are tilted inward by the black hole's gravity.

 

[Matterwave]

Gravity in Newton is described only by one number at each point in space, whereas in GR it is described by 10 numbers at each point in spacetime.

In GR, relative velocities are only defined between objects whose worldlines intersect in spacetime. Let's call that the local velocity. The local velocity of a massive object in freefall never exceeds the speed of light, even as it approaches the singularity.

What goes faster than light in GR? It depends on the definition of "what" and "goes", but there is a sense in which space itself falls faster than light into a black hole.

Man you got to be spending all your spare time on Arxiv cus you always have a paper ready...XD

 

[John15]

Peter thanks for the excellent reply but I now have a slight problem over the definition of force. You say that in GR force is something you can feel, if you are in freefall surely you must be accelerating but at exactly the same rate as the force acting on you whatever an accelerometer shows. If you are standing on the Earth you will feel gravity, a set of scales will tell you how much, you are still accelerating towards the Earth's center but the surface stops you moving. While I can see spacetime being curved by mass can affect things traveling through space I have difficulty in seeing how this curvature operates when you stand on a set of scales.
I can see also how light cones can be used to show the propagation of an event but they are not real, just 2d mathematical constructs, would time cones be a better description. A burst of light propagates in all directions in space but only forward in time.
Hope I do not seem argumentative just trying to get a proper handle on things.

 

[D H]

If you are standing on the Earth you will feel gravity ...

You are not feeling gravity. You are feeling everything but gravity. The Earth is pushing up on your feet due to the normal force. It is this upward force that you feel, not gravity.

Imagine taking a roller coaster ride. A good one will have spots where you feel weightless, or even negative Gs (what feels like "down" is upwards). Really good ones will combine that feeling of weightlessness with a roll centered roughly about your heart so that different parts of your body disagree as to which way is "down". Gravity changes only by a tiny, tiny bit in the hundred meters or so elevation changes during the ride. What you feel changes a whole lot. That's because you don't feel gravity.

... a set of scales will tell you how much

Your bathroom scale also doesn't measure gravity. It is measuring the normal force. If you are standing still and aren't cheating (e.g., reaching up and pushing against on the ceiling, or reaching down and pushing against the vanity), that normal force is nearly equal to the gravitational force acting on your body. If you sit on that bathroom scale when you ride a roller coaster it will give some radically different readings than your gravitational weight.

 

[LaurieAG]

The rule in SR that "nothing goes faster than light" is generalized in GR to "nothing moves outside the light cones"; the light cones are surfaces in spacetime formed by the worldlines of light rays radiating from a particular event.

The main difference between Newtonian, 3D+t Euclidian space, and space time, Minkowski/Lorentzian space, is that Minkowski space requires a transformation that involves the imaginary unit i to convert into Euclidian space.

This is probably best illustrated in the derivation from first principles of the charge on a point from a wire in basic EM field theory in electrical physics. The initial start point is the wire and later a transformation based on an imaginary unit allows us to flip between the center of the wire (and the field, central perspective) and the point where the measurement is being taken (on the edge of the field). The empirical results are real because the measurement is done externally and the measurement could not occur at the physical location of the wire (short circuit of loop, from central perspective not the external perspective).

The external perspective (reality) on the field theory is from an outside point pointing inwards and the internal perspective (imaginary) is from the inside pointing outwards.

Post 19 in the philosophy thread, Overnight everything has doubled in size, shows how a dimensionless constant can be used to forge a consistent observation scale for the comparison of different measurement data sets of the light emitted from rotating sources being observed on galactic year scales in a Euclidian 3D+t space.

The attached image shows the Light paths expected traveling from source(s), rotating around a stationary galactic centre, to a stationary observation point in a Euclidian 3D+t space during one complete rotation. These curved but straight light paths are to be expected at the observation time when physically observing rotating sources continue to rotate and emit while the light being observed travels all the way to the observer at a consistent speed. If we apply an imaginary translation we will get Euclids original internal perspective (imaginary) of a beam of light traveling in a straight line to the stars, when he opened his eyes at night. This internal field theory conception should not be considered as Euclidian space which provides an external perspective (reality).

The closest I can get to modelling an external perspective (reality) in measureable internal perspective (imaginary) reality is in the images attached to the Fun Photos & games thread, Feedback loop photos. The only variables in the loop setup required to create the last image posted is only three different angles of rotation, one distance and a carefully placed mirror. If we remove the mirror we can get a stable perfect orb state, which is obviously constructed of the cuumulative lag within the measuring system, as you can see in the screen captures (the outside spinning and being drawn into the center, i.e the reality to imaginary flip).

A measureable internal perspective (imaginary) reality is not equivalent to an external measureable perspective (reality) and that is the real difference.