Question about Gravity and curvature of space time

[Mike442]

That is basically the Newtonian explanation, with "curvature of space-time" randomly thrown in. In General Relativity there is no "gravitational force", that opposes the force of the hand. That's why the ball experiences proper-acceleration upwards, when held in the hand.

Sorry A. T. but I have to disagree with the statement that the "curvature of space-time is randomly thrown in." Einstein"s field equations reduce down to the Newtonian equation of F = G(M1)(M2)/R^2. So Newton's equation is valid and connected to general relativity. Newton couldn't explain why his equation worked since he developed it from scientific observation and experimentation. He didn't realize that the R squared term in the denominator is actually the result of multiplying the curvature of space-time squared ( 1/R)^2 times G(M1)(M2). Newton's equation works because of the curvature of space-time. I know the conventional why of teaching this equation is that R is the distance between the centers of the two masses M1 and M2. Also, F= ma =G(M1)(M2)/R^2 where a equals the gravitational acceleration constant g.

 

[A.T.]

So Newton's equation is valid and connected to general relativity.

The two models yield similar results in some cases. But the OP asked for an explanation based on general relativity, not on Newtonian force of gravity.

 

[PeterDonis]

the R squared term in the denominator is actually the result of multiplying the curvature of space-time squared ( 1/R)^2

1/R^2 is not the curvature of spacetime squared. (Purely in terms of units, curvature has units of 1 / length^2, not 1/length, so 1/R^2 has the units of curvature, not curvature squared. But just having the right units is not enough.) If we take a 2-sphere at radius R around a gravitating body, 1/R^2 is the intrinsic curvature of that 2-sphere. (Note that R is not the actual physical distance to the center of the body; that distance is larger than R.) But that is not the same as the curvature of spacetime.

Spacetime curvature is given by the Riemann curvature tensor, which is not a single number; it has 20 independent components in general, and in vacuum it has 10. A typical component in the vacuum surrounding a gravitating body is of order $M / r^3$ in geometric units (or $GM / c^2 r^3$ in conventional units), and describes the tidal gravity produced by the body.

 

[Quantumjump]

Hello all

I just joined this forum so forgive me for jumping right in but I have a question about Gravity and the curvature of space time that I can't get answer with a Google search. My question: though I understand that an object remains in orbit because of the curvature of space time and it is this curvature which is responsible for Gravity, but what causes an object that is stationary to fall toward the center of mass if nothing sets it in motion? Does the curvature of space give it a nudge? If so How? Why does a ball which is motionless in my hand fall if I let go of it without giving a push? I understand that if I set it into motion fast enough that it will fall around the Earth following the curvature of space but what makes it move toward center of mass if no force is acted on it?

Hi Christine,

when you release a ball that is about to fall down, its initial direction will make it to follow a particular spacetime curve (=geodesic) that is different of the one that would correspond to an orbit (which, in fact, is a curve associated to another geodesic). It is wrong to speak of "force" because in essence, we are only allowed to speac of matter-energy on one hand, and space-time distorsions of the other hand . There is no "force" and it's better to avoid this word to avoid confusion. HOWEVER, in the approximation of weak tensor fields, it is possible to prove that Einstein's equations reduce to a form with has the structure of Newton's Law. This basically means that Newton's Law F=ma can be seen as a reinterpretation of weak-field limit of Einstein's equations.

 

[looka]

Ah, too late for the party but it clicked with me like this (please someone point if it's wrong).

Not only space is curved but time as well. And time does move forward.
So if curved surface of Earth is space in one dimension and time in other, and we are standing still some distance away at equator, and times keeps pushing us forward, we will 'accelerate' towards each other.

 

[stevendaryl]

Ah, too late for the party but it clicked with me like this (please someone point if it's wrong).

Not only space is curved but time as well. And time does move forward.
So if curved surface of Earth is space in one dimension and time in other, and we are standing still some distance away at equator, and times keeps pushing us forward, we will 'accelerate' towards each other.

Yes, that's a good analogy. Think of forward in time as moving north, and spatial separation is measured as the east-west distance. Then as time goes on, objects move north along a line of longitude, and their east-west distance decreases.