Minkowski space, Light cones and Schwarzschild.

[JohnSSM]

In Minkowski space, it seems that the axis for the dimension of time is represented by the light cone. And it also seems that this dimension is always "aligned" with one of the space dimensions. The observer in the "hypersurface of the present" seems to exist in a 2 dimensional space frame (x,y) that rotates or transforms at perpendicular angles to the observer's motion, which becomes the space frame (z) and the dimension of time. I am basically describing a graphic at wikidpedia found here:

http://en.wikipedia.org/wiki/Minkowski_space

Then I was reading through the Schwarzschild metric info and noticed these 2 terms in the equation.

θ is the colatitude (angle from North, in units of radians),

φ is the longitude (also in radians),

Are those coordinate based terms related to the angle or direction of the light cone in Minkowski's manifold model?

I was also hoping to ask how they figured the colatitude and longitude of any object...Where does one harvest that info and how?

[elfmotat]

The graphic you're describing is just a picture of past and future light-cones. Its relation to the point of your post seems unclear.

 

"Minkowski space" is just a fancy name given to flat spacetime, i.e. spacetime with no curvature/gravity.

 

Co-latitude and longitude are two of the four Schwarzschild coordinates on Schwarzschild spacetime. They mean the same thing that they do in ordinary spherical coordinates. I fail to see how this relates to Minkowski space or light-cones.

[Mordred]

The collateral and lateral terms are polar coordinates. Angular momentum is often done in the colateral coordinate for example.

 

X,y,z,t are Cartesian coordinates.

 

I recommend studying the two coordinate systems in detail and look at the conversions between the two.

 

Inertial frames of reference can use either system so knowing how to convert between the two Is an invaluable skill. This will also help you connect the sine and cos rules as they are involved in the conversions.

Curvature influences are more often than not use the polar coordinate system. (In some cases one assigns the north direction of influence.).sine waves is a good example.

In this case the curve of a sine wave is more convenient in polar coordinate change than doing the coordinate change in Cartesian coordinates.

( spherical objects and interactions are typically done in polar coordinates)

[JohnSSM]

The collateral and lateral terms are polar coordinates. Angular momentum is often done in the colateral coordinate for example.

 

X,y,z,t are Cartesian coordinates.

 

I recommend studying the two coordinate systems in detail and look at the conversions between the two.

 

Inertial frames of reference can use either system so knowing how to convert between the two Is an invaluable skill. This will also help you connect the sine and cos rules as they are involved in the conversions.

Curvature influences are more often than not use the polar coordinate system. (In some cases one assigns the north direction of influence.).sine waves is a good example.

In this case the curve of a sine wave is more convenient in polar coordinate change than doing the coordinate change in Cartesian coordinates.

Yes, how do they determine the north so as to make coordinates based on it?

 

I suppose another way to ask my question is "Are the polar coordinates used to define the schwartszchild metric, on the same axis as motion (the light cone) in the Minkowski model?"

[Mordred]

Linear vectors are more often done in Cartesian. Ie a constant force applied in a vector direction.

Yes, how do they determine the north so as to make coordinates based on it?

 

I suppose another way to ask my question is "Are the polar coordinates used to define the schwartszchild metric, on the same axis as motion (the light cone) in the Minkowski model?"

Look at the metric to describe the flat euclidean plane. Positive x is typically north. - X south +y east -y west

A helpful visualization tool Is look at how polar coordinates is used to map locations on the Earth.( Polar coordinates is common in navigation)

We could map locations on Earth in Cartesian coordinates but it's more inconvenient to do so.

Here is a basic polar to Cartesian coordinate conversion link

 

http://www.mathsisfun.com/polar-cartesian-coordinates.html

[JohnSSM]

You may not understand the question...the ole vagueness at work, no doubt...

I was looking at this equation, imagining that I wanted to solve it myself...

where

is the proper time (time measured by a clock moving along the same world line with the test particle),

c is the speed of light,

t is the time coordinate (measured by a stationary clock located infinitely far from the massive body),

is the radial coordinate (measured as the circumference, divided by 2π, of a sphere centered around the massive body),

θ is the colatitude (angle from North, in units of radians),

φ is the longitude (also in radians), and

is the Schwarzschild radius of the massive body, a scale factor which is related to its mass M by r_s = 2GM/c², where G is the gravitational constant.

<sup>I see that I need 2 sets of data for time, and I also see that I need 2 sets of data for "polar postion" or "axis" ...

So, if i were attempting to solve the schwarszchild metric where do I get the info to plug into the colatitude and longitude? These measurements assume an axis has been established...How do we establish that axis to make the measurement for colatitutde and longitude?</sup>

I recommend studying the two coordinate systems in detail and look at the conversions between the two.

Were sine and cosine developed strictly for transforming polar coordinates to cartesian coordinates?

Edited January 21, 2015 by JohnSSM

[Mordred]

You need to apply the polar to Cartesian conversions I've posted. See the math is fun link the conversion rules will answer your question. ( when do you choose to use sine,sineh cos and Cosh?). See link.

You may not understand the question...the ole vagueness at work, no doubt...

 

I was looking at this equation, imagining that I wanted to solve it myself...

 

where

is the proper time (time measured by a clock moving along the same world line with the test particle),

c is the speed of light,

t is the time coordinate (measured by a stationary clock located infinitely far from the massive body),

is the radial coordinate (measured as the circumference, divided by 2π, of a sphere centered around the massive body),

θ is the colatitude (angle from North, in units of radians),

φ is the longitude (also in radians), and

is the Schwarzschild radius of the massive body, a scale factor which is related to its mass M by r_s = 2GM/c², where G is the gravitational constant.

 

^{I see that I need 2 sets of data for time, and I also see that I need 2 sets of data for "polar postion" or "axis" ...}

 

So, if i were attempting to solve the schwarszchild metric where do I get the info to plug into the colatitude and longitude? These measurements assume an axis has been established...How do we establish that axis to make the measurement for colatitutde and longitude?

Were sine and cosine developed strictly for transforming polar coordinates to cartesian coordinates?

Now your getting it.

[JohnSSM]

Here is a basic polar to Cartesian coordinate conversion link

 

http://www.mathsisfun.com/polar-cartesian-coordinates.html

That really got to the point quickly and clearly...nice link...

You need to apply the polar to Cartesian conversions I've posted. See the math is fun link the conversion rules will answer your question. ( when do you choose to use sine,sineh cos and Cosh?). See link.

 

Now your getting it.

Apply them to what though? At some point I need some actual data to manipulate to get the 2 space coordinates to solve the equation...

 

Next, Ill ask how Im supposed to get the time off of a clock thats inifinite distance away...

I also noticed, they dont transform perfectly between polar and cartesian...doesnt that bother math folk?

All Tangence does is take an angle and give it a ratio? and back again? Why could no one ever explain that in high school?

[Mordred]

Use the rules on your Schwartzchild image, practice the conversions so you don't need to think about them. Start with the examples on that link.

 

In your other thread I posted a calculus article. There is examples there.

 

Its best to eat the Apple one bite at a time.

 

As far as exact conversions, there is more exacting coordinate systems. Though in the angles they get extremely precise. Degrees minutes seconds for example.

To be honest I can't recall a single formula that uses tan

[JohnSSM]

Use the rules on your Schwartzchild image, practice the conversions so you don't need to think about them. Start with the examples on that link.

 

In your other thread I posted a calculus article. There is examples there.

 

Its best to eat the Apple one bite at a time.

 

As far as exact conversions, there is more exacting coordinate systems. Though in the angles they get extremely precise. Degrees minutes seconds for example.

To be honest I can't recall a single formula that uses tan

Calculators do all that stuff easy...what conversions are you referring to?

 

I cant do the math until I have the values to plug in...

And also, what is "d"? They dont define it in the description...lots of times, the biggest difficulty for me is knowing what the terms stand for...

Man...what is this? Looks amazing

 

http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

[Mordred]

Distance calculators is no replacement to knowledge. Particularly when you need to adapt a formula

[JohnSSM]

Distance calculators is no replacement to knowledge. Particularly when you need to adapt a formula

I allready see how the relationship works for the transformation...I totally get it...

[Mordred]

As to the last calculator that's the lightcone calculator for the expansion history of the universe. I helped write the user guide and tutorial. Jorrie is the programmer. It can use the FLRW metric to show any distance measures in Cosmology expansion. As well as plot it in the lightcones and world lines. Starts at CMB forward and can plot 88 billion years into the future.

[JohnSSM]

What I see are objects of mass "surrounded" by their own polar coordinates and curves in 3d, and they all reside within a larger cartesian coordinate system based on outside influences...but the polar must merge with the cartesian to truly connect the object and its own forces to the outside forces acting upon it...vague?

[Mordred]

Works with both WMAP and Planck datasets

[JohnSSM]

As to the last calculator that's the lightcone calculator for the expansion history of the universe. I helped write the user guide and tutorial. Jorrie is the programmer. It can use the FLRW metric to show any distance measures in Cosmology expansion. As well as plot it in the lightcones and world lines. Starts at CMB forward and can plot 88 billion years into the future.

It gives you the lightcone and world line for the entire universe as a whole?

[Mordred]

What I see are objects of mass "surrounded" by their own polar coordinates and curves in 3d, and they all reside within a larger cartesian coordinate system based on outside influences...but the polar must merge with the cartesian to truly connect the object and its own forces to the outside forces acting upon it...vague?

Basically accurate all forms of influence can be described by geometry

It gives you the lightcone and world line for the entire universe as a whole?

Correct

[JohnSSM]

Basically accurate all forms of influence can be described by geometry

 

Correct

Basically accurate is the firmest rating of accuracy you have ever given me...blush

[Mordred]

Roflmao you are coming a long way since your first post

[JohnSSM]

Adopting that view has allowed to me see the existence of space-time as separate from the objects within it....as if space-time were a cartesian like field, and objects created polarized fields within it...

[Mordred]

As long as you can define the coordinate relations that is in my opinion accurate. All models are described by mathematics. Most of the complex terminology in peer reviewed papers are differential geometry descriptives. Look on math is fun for transformations. Ie rotation transformation.

Ps this also describes symmetry

Including symmetry in particle physics

Double PS that includes QM, fields, and string theory

 

 

There is one expression to take note.

 

"The universe does not care how we measure it"

 

I'll let you ponder upon its philosophical ramifications.

Edited January 21, 2015 by Mordred

[JohnSSM]

Ive been looking into rotation transformation since my first post...Finding that the LT was essentially a linear transformation with rotational effects...learning about Boosts and how rotation decreases boost...

I have a question about symmetry and spheres in schwarzschild that isnt clear to me from the info ive been reading...I assume they are referring to symmetry of shape, but are they also referring to symmetry of density? Or, does shape really even matter as long as it is determined to be symmetrically dense? I can imagine a symmetrically spherical object, but would it make a difference if one half was gold and the other was sodium? Im trying to create an example where the object were perfectly spherical, but not evenly distributed within..."off balance" so to say...

IS schwarzschild only concerned with shape symmetry? Or does it also imply a symmetrical distribution within?

There is one expression to take note.

"The universe does not care how we measure it"

I'll let you ponder upon its philosophical ramifications.

Are you saying that we can define what is north? I would think each object would dictate its own polarity, possibly based on the distribution of material and what kind of spin it causes unto itself...spin seems to create poles....

[Mordred]

Symmetry is the relation being modelled. This can be shape, density, pressure, or interaction. Etc

 

We define how we choose to model any system and how we describe that system. Naturally we tend to use easily understood reference points and baseline values as a reference point.

 

North is convenient as we normally graph x as longitudinal, y as horizontal. For baselines we look for most common average value or zero then model the deviations from that baseline. ( good example temperature change from room temperature, then model the change in temperature from that point)

 

You can always set the initial values,then describe the deviations.

 

Most often this is based on convention and logic,

Edited January 21, 2015 by Mordred

[JohnSSM]

Symmetry is the relation being modelled. This can be shape, density, pressure, or interaction. Etc

 

We define how we choose to model any system and how we describe that system. Naturally we tend to use easily understood reference points and baseline values as a reference point.

 

North is convenient as we normally graph x as longitudinal, y as horizontal. For baselines we look for most common average value or zero then model the deviations from that baseline. ( good example temperature change from room temperature, then model the change in temperature from that point)

 

You can always set the initial values,then describe the deviations.

 

Most often this is based on convention and logic,

So the schwarschild metric doesnt nessecarily need a symmetrically spherical object like a perfectly round ball?...the symmetry that seems to matter more is the density, caused by the pressure, caused by the interactions...you could actually end up with all sorts of shapes if you were allowed to change the internal density in certain parts of the object....but if you dont, and keep it densely symmetrical, a sphere is what you end up with...From what ive gathered, anything other than 3 dimensional, density symmetry and the object will spin under its own influence...and then schwarszchild no longer can model the object...

 

IS there a model that incorporates non symmetrical objects with spin? And how about adding motion? and other objects?

I believe I have found the answer I sought whilst reading on mass quadrupoles...

 

"The mass quadrupole is analogous to the electric charge quadrupole, where the charge density is simply replaced by the mass density and a negative sign is added because the masses are always positive and the force is attractive."

 

So it doesnt take 4 magnets to create this quadrupole of mass...it takes the four dimensions that the mass occupies in space-time...and then you can figure your gravitational quadrupole and how that object effects space time locally, starting at its own center of mass...