General Doubts about Space Time

[gaugeboson]

Here are a few questions about space-time.
1. Mass is directly proportional to curve/dent in s-t. (space-time). What is the constant of proportionality? So, by 1kg. mass how much of a curve is produced? OR by how much mass does the pit deepen by 1 unit length?
2. By critical velocity and centrifugal force, I can understand why planets remain in orbit. But if Sun creates a bowl or a cone shaped pit in space-time, why don’t the planets cave in onto the Sun?
3. A black hole is crushed by internal gravity into a singularity. Just before it becomes a singularity, its volume must be less than an atom’s. Thus there would be millions of protons where there should have been just 1. So, how is matter arranged then? What is this glob of matter called? What is its state? (it is definitely not like particles)
4. Space-time can explain why planets travel when nearer to the Sun. But suppose a rocket is going directly towards the Sun, it should take more time than linear travel (as it also has to travel through the curves of s-t.). But if time goes slower near the Sun, we feel as if it is taking the same time. Is this logical?

 

[gaugeboson]

I'm only a student so please bear with these doubts. Hope you don't get bored reading them!

 

[chronon]

1. Mass is directly proportional to curve/dent in s-t. (space-time). What is the constant of proportionality? So, by 1kg. mass how much of a curve is produced? OR by how much mass does the pit deepen by 1 unit length?

The popular picture of a mass making a dent in space isn't really correct. The relationship of spacetime curvature to mass is given by Einstein's equation of general relativity.

2. By critical velocity and centrifugal force, I can understand why planets remain in orbit. But if Sun creates a bowl or a cone shaped pit in space-time, why don’t the planets cave in onto the Sun?

Even in the incorrect view of a marble in a bowl, the marble will keep going round forever if there's no friction.

3. A black hole is crushed by internal gravity into a singularity. Just before it becomes a singularity, its volume must be less than an atom’s. Thus there would be millions of protons where there should have been just 1. So, how is matter arranged then? What is this glob of matter called? What is its state? (it is definitely not like particles)

If you're talking about what happens inside the event horizon, then I would say that concepts like 'just before' don't really apply.

4. Space-time can explain why planets travel when nearer to the Sun. But suppose a rocket is going directly towards the Sun, it should take more time than linear travel (as it also has to travel through the curves of s-t.). But if time goes slower near the Sun, we feel as if it is taking the same time. Is this logical?

For a rocket going between two events in spacetime, the time experienced (proper time of the rocket) will be longest if it is in freefall. Any acceleration will shorten the proper time.

 

[pervect]

Here are a few questions about space-time.
1. Mass is directly proportional to curve/dent in s-t. (space-time). What is the constant of proportionality? So, by 1kg. mass how much of a curve is produced? OR by how much mass does the pit deepen by 1 unit length?

This is given by Einstein's equation

G_uv = 8 Pi T_uv

G_uv measures the curvature. T_uv measures the mass. Actually, T_uv measures the mass density, not the mass - so curvature is proportional to density, not the mass itself. The mass density is also just one component of T_uv (which is a tensor, think of it as a 4x4 matrix if you like) so it determines only one component of the curvature, G_uv. Momentum and pressure are some of the other components of T_uv which affect other components of the curvature tensor.

Note that it's not space that is curved, but space-time.

2. By critical velocity and centrifugal force, I can understand why planets remain in orbit. But if Sun creates a bowl or a cone shaped pit in space-time, why don’t the planets cave in onto the Sun?

I'm not sure why you think they should "cave intio the sun". I suspect you're pushing the analogy too hard, and also confusing space with space-time. (The usual rubber sheet anaologies perpetrate this confusion). It's admitittedly a bit difficult to visualize a curvature in time, the best approach is to make a space-time diagram which represents the time dimension as spatial dimension, and then think about the diagram being drawn on a curved surface. The concept here that's most useful is one of "Geodesic deviation", unfortunately I don't have any really good URL's for this on a basic level.

3. A black hole is crushed by internal gravity into a singularity. Just before it becomes a singularity, its volume must be less than an atom’s. Thus there would be millions of protons where there should have been just 1. So, how is matter arranged then? What is this glob of matter called? What is its state? (it is definitely not like particles)

Nobody knows, thouth quantum gravity experts may have theories. It's outside the scope of GR itself, though.

4. Space-time can explain why planets travel when nearer to the Sun. But suppose a rocket is going directly towards the Sun, it should take more time than linear travel (as it also has to travel through the curves of s-t.). But if time goes slower near the Sun, we feel as if it is taking the same time. Is this logical?

This is confusingly worded, I think, but not totally misguided. It would be supportable to say, for instance, that a rocket falling into a black hole will take an infinite amount of coordinate time to reach the event horizon, but only a finite amount of time will elapse on the clocks attached to the rocket (i.e. it will take only a finite amount of proper time).

If you are interested in the orbits and behavior of objects in relativistic gravity, there are some good web applets out there, such as

http://www.fourmilab.ch/gravitation/orbits/

 

[pmb_phy]

This is given by Einstein's equation

G_uv = 8 Pi T_uv

G_uv measures the curvature.

Since when? G_uv is composed of the Ricci scalar and Ricci tensor. Its quite possible that G_uv can be zero and yet have non-zero spacetime curvature. A perfect example is the Schwarzschild spacetime.

Pete