How Does Mass Influence the Curvature of Space-Time?

[Muon12]

This is a question I'm throwing out there for anyone who may know: Is there an exact known ratio for the curvature of space-time in relation to an object's mass? In other words, is there a value or equation that is used to determine how relatively distorted space will be due to an object's gravitational pull? And what kind of units are used to describe 4-dimensional space, or higher for that matter (what can I say, I like asking questions)?

After all, ignorance is not always bliss...8)

 

[selfAdjoint]

The expression you are thinking of is called the Schwartzschild metric. It describes the curvature of spacetime (not just space) in the vicinity of a gravitating mass. This works if the mass is not rotating and doesn't have an electric charge. There are other metrics that describe the curvature in those cases.

Usually in relativity the spatial components are measured in lengths of some kind, and the time component is -- also measured in length. They multiply its time units by a speed - length unit per time unit - to convert it. The speed they use is the speed of light.

 

[yogi]

There is a simple formula for a spherical mass M - the change in the radius (delta r) = MG/3c^2 For the Earth this turn out to be about 1.5 millimeteres

 

[photon]

density

Doesn't the body's density also affect how the body affects space time?
From what I understand, something can have a low mass, high density, and therefore make a huge "pit" in space-time.[b(]

 

[Coughlan]

yea but what about those strings? i think that is the name, its the stuff left over from the big bang that is super dense like the densest matter can get...Wouldn't that create a rather large hole...help me out here

 

[chroot]

Originally posted by photon
Doesn't the body's density also affect how the body affects space time?
From what I understand, something can have a low mass, high density, and therefore make a huge "pit" in space-time.[b(]

Yes, density affects the so called "surface gravity." Cosmic strings (if they exist) are infinitely thin yet very massive, and thus would have surface gravity tending towards infinite.

- Warren

 

[Coughlan]

so would that make time infinite around the object or on the object? how does that work?

 

[yogi]

We can always consider a massive object as a point as did Newton - and from a computational standpoint consider all the mass centralized - thus the gravity of an object such as the Earth is no different at a distance 10,000 miles from its center than it would be if it were compressed to the size of a black hole - density only comes into play if you are making calculations on the surface and you know the radius and the total mass - but as to its effect upon external matter - all spherical objects can be regarded simply as a point with a mass-energy concentrated in a singularity.

If you consider the universe as a whole - it has the correct mass for a black hole having a size equal to the Hubbe radius. Again - the affect upon space is:

delta r = MG/3c^2 - nothing to do with density per se

 

[marcus]

Originally posted by selfAdjoint
The expression you are thinking of is called the Schwartzschild metric. It describes the curvature of spacetime (not just space) in the vicinity of a gravitating mass. This works if the mass is not rotating and doesn't have an electric charge. There are other metrics that describe the curvature in those cases.
...

selfAdjoint and Yogi have already answered this question. I can add a concrete example to illustrate.

Muon's question is: "Is there an exact known ratio for the curvature of space-time in relation to an object's mass? In other words, is there a value or equation that is used to determine how relatively distorted space will be due to an object's gravitational pull?"

Well one way of measuring how distorted space is to say how much a ray of light will be bent as it passes a massive object.

It depends on the mass and on how close the ray of light comes to the mass, and a formula rather like the one that Yogi wrote can be used

If something has mass M then a handy length to calculate (if you want to say how much it bends space) is GM/c².
For the mass of the sun it comes to 1.5 kilometers or a bit less than a mile.

you can calculate it yourself by looking up in a book to find the mass of the sun in kilograms and the speed of light in meters per second and the constant G (in kilograms, meters, and seconds). A lot of stuff cancels and it boils down to 1.5 kilometers.

The angle a ray is bent when it passes within distance D of the (center of the) sun is some very small number of radians---a tiny fraction of a radian actually---which you calculate very easily. It is just 4 x this length divided by D.

So since the length is 1.5 kilometer, 4 x the length is 6 kilometer.

And if the light passes within 6 million kilometers of the sun, then the angle is just 6 divided by 6 million-------a millionth of a radian.


If the light passes within 800,000 km of the sun, then the angle of bending is just 6 divided by 800,000-----6/8 of one 100,000th of a radian.