Visualizing Space Curvature

[Lazarus]

It takes 4 dimensions to describe the curvature of 3 dimensional space, (w,x,y,z). Since the curvature of space is equivalent to gravity, w is proportionate to 1/d^2 where d is the distance from a mass.

 

To visualize a straight line affected by the curvature of space we can set y and z to zero and use a line parallel to the x axis and the closest distance to the mass set to one. Next calculate the value of w at points for some values of x and plot w against x.

 

The equation to calculate the points on the line is w = (x^2 +1)^.5.

 

The points calculated are:

x d w

-5 5.0990 0.0385

-4 4.1231 0.0588

-3 3.1623 0.1000

-2 2.2361 0.2000

-1 1.4142 0.5000

0 1.0000 1.0000

1 1.4142 0.5000

2 2.2361 0.2000

3 3.1623 0.1000

4 4.1231 0.0588

5 5.0990 0.0385

 

 

Is this correct?

[Strange]

It takes 4 dimensions to describe the curvature of 3 dimensional space

 

No it doesn't. It can be described as intrinsic curvature, which do not require it to be embedded in a higher dimensional space: https://en.wikipedia.org/wiki/Curvature#Higher_dimensions:_Curvature_of_space

[Lazarus]

 

No it doesn't. It can be described as intrinsic curvature, which do not require it to be embedded in a higher dimensional space: https://en.wikipedia.org/wiki/Curvature#Higher_dimensions:_Curvature_of_space

Thank you for the link but it didn't really help me much.

 

So we can use 5 dimensions to describe 4 dimensional space/time. Set t, y and z to zero. That should allow a depiction of the w, x slice.

 

Didn't Einstein originally consider only space curvature to calculate the bending of light by the sun, then doubled his result by adding the effect of time curvature?

[Mordred]

No geodesics are rather tricky to derive but the related formulas are here.

 

https://en.m.wikipedia.org/wiki/Geodesics_in_general_relativity

 

Should be noted there is often more than 1 geodesic. Photons follow null geodesics.

Edited April 1, 2016 by Mordred

[Strange]

So we can use 5 dimensions to describe 4 dimensional space/time. Set t, y and z to zero. That should allow a depiction of the w, x slice.

 

No it is modelled in four dimensions.

[ajb]

So we can use 5 dimensions to describe 4 dimensional space/time.

Yes, but you do not need to... that is the point.

 

EDIT: And it can be misleading as what you get depends on how you present the lower dimensional space in the higher one. A question, is a cylinder flat or curved?

Edited April 1, 2016 by ajb

[geordief]

Are there any representations using 3D as in films like Avatar that make a better job of displaying the way space time is curved in the presence of mass?

 

Also is it correct to say that ,when we open our eyes every morning we are in fact looking at the world in 4D (or 3d+1) dimensions -just that speeds are very low ?

Edited April 1, 2016 by geordief

[ajb]

Are there any representations using 3D as in films like Avatar that make a better job of displaying the way space time is curved in the presence of mass?

Not that I am aware of. I am not sure what can be gained in this way.

[Lazarus]

To compare a line in flat space and the line bent by space/time curvature, consider a line in flat space that is tangent to the sun’s surface. The same line in curved space/time (the geodesic) will be bent and stretched near the sun. A light year from the sun in each direction, the curvature of space/time from the sun’s gravity would be infinitesimal so the flat space line would approximant the curved space/time line.

 

This is inconstant with the concept of the sun changing the direction of light because of space/time curvature.

 

What is going on?

[Mordred]

Your understanding is wrong. At sufficient distance from mass. The geodesic is a straight line. Read your post again and find the line in error.

[Strange]

The same line in curved space/time (the geodesic) will be bent and stretched near the sun.

 

That is why the direction of the light is changed.

 

 

A light year from the sun in each direction, the curvature of space/time from the sun’s gravity would be infinitesimal so the flat space line would approximant the curved space/time line.

 

And a light year from the Sun there would be negligible effect on light.

[swansont]

To compare a line in flat space and the line bent by space/time curvature, consider a line in flat space that is tangent to the sun’s surface. The same line in curved space/time (the geodesic) will be bent and stretched near the sun. A light year from the sun in each direction, the curvature of space/time from the sun’s gravity would be infinitesimal so the flat space line would approximant the curved space/time line.

 

This is inconstant with the concept of the sun changing the direction of light because of space/time curvature.

 

What is going on?

On the contrary, to me that sounds exactly like the scenario where a star that's behind the sun is still visible. The line from the star to the sun is straight, but near the sun there is a perceptible bend, and further away toward us it's straight again. So if one follows it, it points to a region behind the sun.

[Lazarus]

On the contrary, to me that sounds exactly like the scenario where a star that's behind the sun is still visible. The line from the star to the sun is straight, but near the sun there is a perceptible bend, and further away toward us it's straight again. So if one follows it, it points to a region behind the sun.

 

Excellent explanation, thanks.

 

I do have a small problem with it. The distance from the sun to the earth is about 220 times the radius of the sun. The space curvature should be quite small at near the earth so the geodesic should be pretty close to the flat space straight line. The direction of motion of the light should be differenr from the effect of gravity.

 

[Strange]

The first of your diagrams is much closer to reality.

[Lazarus]

The first of your diagrams is much closer to reality.

 

 

Yes. That is the easy part.

We can fix the direction the light arrives at earth by increasing the space/time curvature so the line passes the sun at a distance a bit greater than the radius of the sun. This seems to correlate to the Einstein Rings around galaxies since the rings appear to be greater than the radius of the galaxy.

 

However, that still leaves me with the problem that gravity changes the direction of light permanently but the space/time curvature has to return to the original direction.

[Mordred]

Just like light being bent in a prism. It gets bent in the prism. Once it exits the prism light will continue in a straight line. There are numerous classical examples without needing spacetime curvature.

 

Many mediums have similar effects.

 

In point of detail playing around with light being refracted in a medium is a good exercise to visualize curvature and subsequently the mathematical relations to geometry.

Shine a laser through various shapes of glass. You'd be surprised how much easier it is to understand spacetime curvature.

 

 

Though don't think of spacetime as being it's own medium. The medium like similarities is the geometry changes. Not a substance

Edited April 8, 2016 by Mordred

[StringJunky]

Just like light being bent in a prism. It gets bent in the prism. Once it exits the prism light will continue in a straight line. There are numerous classical examples without needing spacetime curvature.

 

Many mediums have similar effects.

 

In point of detail playing around with light being refracted in a medium is a good exercise to visualize curvature and subsequently the mathematical relations to geometry.

Shine a laser through various shapes of glass. You'd be surprised how much easier it is to understand spacetime curvature.

 

 

Though don't think of spacetime as being it's own medium. The medium like similarities is the geometry changes. Not a substance

I visualise gravity/spacetime curvature as a 360 density gradient, changing consistently with that generated by test masses and their interactions, as described by the appropriate equations.

Edited April 8, 2016 by StringJunky

[Mordred]

I visualise gravity/spacetime curvature as a 360 density gradient, changing consistently with that generated by test masses and their interactions, as described by the appropriate equations.

That works, some of the relations are similar. For example pressure can cause an increase in mass density.

 

As long as it's clear spacetime isn't its own material or substance. However it's filled with the standard model particles of various densities.

 

The trick that helped me to understand Mass. I looked at the meaning of mass "resistance to Inertia.

 

Then I studied how a particle interacts with a field. Ie electromagnetic, weak and strong fields.

 

If mass is resistance to inertia then mass can be gained via a particles interaction via the fields it can interact with. A portion of a particles mass is from the strong, electromagnetic,Higgs and in careful circumstances the weak fields. (Via Higgs)

 

These fields geometry curves due to relativity.

 

Collectively one can consider a gravitational field as the combination of all other fields.

 

You never see this described in this manner, as most relativity textbooks want you to focus on the geometry aspects of mass density influence on the gravitational field. The gravitational field tends to focus on how mass interacts with mass. ( it covers massless ( rest mass) particles in a different geodesic. ( null geodesic. ( ever wonder why the term "null" ?)

 

All the other fields has a mediator particle. For gravity we haven't found the mediator graviton.

 

Yet we don't really need one. Every other field curves in precisely the same relations.

 

So one can consider spacetime geometry to be a representation of all fields combined.

 

Time being a coordinate vector.

 

Also keep in mind you can include a vector or momentum field with the above.

 

( to handle inertial mass)

 

The above is particularly useful to understand why a photon curves due to Geometry. The only field the photon itself interacts with is to be the mediator gauge boson for the electromagnetic field.

 

It doesn't interact directly with gravity in the same manner as particles with mass. However it does interact with fields that also curve due to mass.

( keep in mind every field has its own interaction rules)

**Then of course a particle is essentially an excitation in a field.** that excitation has wavelike and point like properties.

 

Gives one food for thought...

 

For example light travelling through glass. The most dominant source of resistance is the electromagnetic force. ( or field).

 

As all fields can be be modelled as mass density, the mathematics are near identical as light travelling through spacetime. ( The time dimension being tricky to model in a medium) in a loose sense relates to why light slows down in a medium. ( Ie number of interactions to transverse a length path)

Example take a particle carrying say nothing more than momentum ( ie quasi particle).

 

To transmit that momentum through a medium it must mediate the momentum through each particle in its path.

 

A field isn't any different. We map each mediator particle to a coordinate. We map how a property transmits through that field.

 

Mathematically it has the same relations as a medium.

Edited April 8, 2016 by Mordred

[StringJunky]

That works, some of the relations are similar. For example pressure can cause an increase in mass density.

 

As long as it's clear spacetime isn't its own material or substance. However it's filled with the standard model particles of various densities.

 

The trick that helped me to understand Mass. I looked at the meaning of mass "resistance to Inertia.

 

Then I studied how a particle interacts with a field. Ie electromagnetic, weak and strong fields.

 

If mass is resistance to inertia then mass can be gained via a particles interaction via the fields it can interact with. A portion of a particles mass is from the strong, electromagnetic,Higgs and in careful circumstances the weak fields. (Via Higgs)

 

These fields geometry curves due to relativity.

 

Collectively one can consider a gravitational field as the combination of all other fields.

 

You never see this described in this manner, as most relativity textbooks want you to focus on the geometry aspects of mass density influence on the gravitational field. The gravitational field tends to focus on how mass interacts with mass. ( it covers massless ( rest mass) particles in a different geodesic. ( null geodesic. ( ever wonder why the term "null" ?)

 

All the other fields has a mediator particle. For gravity we haven't found the mediator graviton.

 

Yet we don't really need one. Every other field curves in precisely the same relations.

 

So one can consider spacetime geometry to be a representation of all fields combined.

 

Time being a coordinate vector.

 

Also keep in mind you can include a vector or momentum field with the above.

 

( to handle inertial mass)

 

The above is particularly useful to understand why a photon curves due to Geometry. The only field the photon itself interacts with is to be the mediator gauge boson for the electromagnetic field.

 

It doesn't interact directly with gravity in the same manner as particles with mass. However it does interact with fields that also curve due to mass.

( keep in mind every field has its own interaction rules)

**Then of course a particle is essentially an excitation in a field.** that excitation has wavelike and point like properties.

 

Gives one food for thought...

 

For example light travelling through glass. The most dominant source of resistance is the electromagnetic force. ( or field).

 

As all fields can be be modelled as mass density, the mathematics are near identical as light travelling through spacetime. ( The time dimension being tricky to model in a medium) in a loose sense relates to why light slows down in a medium. ( Ie number of interactions to transverse a length path)

Example take a particle carrying say nothing more than momentum ( ie quasi particle).

 

To transmit that momentum through a medium it must mediate the momentum through each particle in its path.

 

A field isn't any different. We map each mediator particle to a coordinate. We map how a property transmits through that field.

 

Mathematically it has the same relations as a medium.

(Response to bolded). Absolutely Mordred, the change in pictorial pixel density with distance, in such an image I have in mind, is as a graphic (as in 'graph-like') representation of the change in gravitational force with distance. It has no nature of 'substance', as commonly understood.

 

That 'a particle is an excitaion in a field', I picked up from you a while ago, and it is now a part of my understanding.

 

Could we say the idea of 'solid' is one of electromagnetic resistance between two bodies?

 

Thanks for the other info, Mordred. I shall absorb what I can.

Edited April 8, 2016 by StringJunky

[swansont]

 

Excellent explanation, thanks.

 

I do have a small problem with it. The distance from the sun to the earth is about 220 times the radius of the sun. The space curvature should be quite small at near the earth so the geodesic should be pretty close to the flat space straight line. The direction of motion of the light should be differenr from the effect of gravity.

 

flatvsst.JPG

 

 

In the second picture you have light bending toward the sun as it passes by (it guides to the right) which is conceptually OK (though exaggerated), but the problem is that at the beginning and end of this you have light bending in the opposite direction (to the left) and there is no reason for it to do that. Light that is incident on the sun is going to hit the sun. We don't see the star superimposed on the sun. We see it along our line-of-sight just off the edge of the sun.

[Lazarus]

 

 

In the second picture you have light bending toward the sun as it passes by (it guides to the right) which is conceptually OK (though exaggerated), but the problem is that at the beginning and end of this you have light bending in the opposite direction (to the left) and there is no reason for it to do that. Light that is incident on the sun is going to hit the sun. We don't see the star superimposed on the sun. We see it along our line-of-sight just off the edge of the sun.

 

If space/time bends down as light approaches the sun, it has to bend up as the light leaves the sun.

 

I thought that we were in agreement that a light year from the sun in either direction that the space/time geodesic is almost the same as the straight line in flat space.

[swansont]

 

If space/time bends down as light approaches the sun, it has to bend up as the light leaves the sun.

 

I thought that we were in agreement that a light year from the sun in either direction that the space/time geodesic is almost the same as the straight line in flat space.

 

 

No, if you go around a bend in the road, you only turn in one direction. Your return to a straight line is a reduction in the amount of your turn, but the wheel never goes past center in the opposite direction. Similarly, the curvature only pulls the light toward the sun, by varying amounts. It never pushes it away.

[Lazarus]

 

 

No, if you go around a bend in the road, you only turn in one direction. Your return to a straight line is a reduction in the amount of your turn, but the wheel never goes past center in the opposite direction. Similarly, the curvature only pulls the light toward the sun, by varying amounts. It never pushes it away.

 

If the road has a dip in it then the road goes down but comes back up.

[Mordred]

 

Could we say the idea of 'solid' is one of electromagnetic resistance between two bodies?

 

 

The majority of the mass in a solid is the binding energy of the electromagnetic force. Its so dominant it's a good approximation. It would be more accurate to think the field resistance. If you take the term binding energy as a field.

 

Rather than two bodies. This will help as a field is modelled via coordinates in GR. So the relations to a particle from one coordinate to another will vary.

 

Just a side note often people think of solids as having unique properties to a gas.

 

The only real difference is density relations. It's perfectly valid to apply the ideal gas laws with a solid for example.

However you have to be careful in the gas law definitions. For example you probably wouldn't treat a solid as an adiabatic fluid. An adiabatic fluid has no net flow of energy to its surrounding where a solid can lose temperature to its surroundings.

 

So in your example if you want to model the interactions between two solids.

 

You know from the above the total rest mass of each solid is due to total confinement energy/internal resistance to inertia.

 

So mathematically in terms of fields its simpler to separate each solid and calculate each seperate. Then model the influence of one solid to another via the background field.

 

This essentially is already being done via mass to mass relations between the two objects.

 

If the road has a dip in it then the road goes down but comes back up.

A road can go anywhere you want it to.

 

A photon follows a straight path, however a straight path can be curved

Due to geometry change

Edited April 8, 2016 by Mordred

[swansont]

 

If the road has a dip in it then the road goes down but comes back up.

 

 

That's not what's going on here, as I explained. The star is not seen as being in line with the sun. So this analogy fails.