An Alternative to a singularity?

[Jeffocal]

But, time dilation is a function of gravitational potential and the person at the center of the star has greater potential [where potential (U) is considered positive] than the person on the surface. The further from the center of the star, the less the potential and the

faster

clocks run.

rather than slower.

~modest

 

Ya but I believe that is only true if you are observing it from the perspective of someone who is outside of stars surface. However relativity says that the laws of physics must be the same for all observers. If you assume the person at the center of the star has relativity greater potential than the person on its surface I think to keep the laws the same for both an observer who is at its center as for one on it surface one would have to say that an observer on the surface of a star would have a relativity greater potential than one at the center.

 

It appears to me many have misused or misunderstood Einstein's basic concepts to justify the existence of things like an event horizon.

[modest]

Everyone can use the same physics (general relativity) and everyone concludes that the clock on the surface runs faster than the clock at the center using that physics—including the people in those positions.

 

The laws of physics being the same for everyone does not mean everyone will have the same potential any more than it would mean everyone feels the same acceleration or everyone is the same distance from some given mass. The physics that everyone uses can be considered:

 

[math]T = \frac{T_0}{\sqrt{1-U/c^2}}[/math]

 

while different observers have different values of U.

 

I do, however, agree with you to an extent about the event horizon. Because Schwarzschild was the first to make an exact solution to GR and his metric broke down at the event horizon many people took the horizon to be a kind of physical boundary in space. This is not really the case with GR, but just an effect of Schwarzschild coordinates. In other words: GR doesn't break down at the horizon, Schwarzschild's metric does (because it's static).

 

~modest

[Pluto]

G'day from the land of ozzzz

 

This paper maybe of interest to the topic. The ABS is explains the point without further writting.

 

[astro-ph/9910408] Non-occurrence of trapped surfaces and Black Holes in spherical gravitational collapse: An abridged version

Non-occurrence of trapped surfaces and Black Holes in spherical gravitational collapse: An abridged version

 

Authors: Abhas Mitra (BARC, Theory Division)

(Submitted on 22 Oct 1999 (v1), last revised 5 Dec 2000 (this version, v5))

 

Abstract: We have shown in that for arbitrary EOS and radiation transport properties, (even) the idealized spherical gravitational collapse DOES NOT lead to the formation of trapped surfaces: 2GM(r,t)/R <=1. Hence all singularity theorems of Hawking, Penrose and Geroch, built on the assumption of formation of trapped surfaces, get invalidated! And this inequality, demands that M->0 if indeed R->0. We have shown that the final state corresponds to a zero mass BH state and, this state would occur only after infinite proper time indicating that GR is indeed the only naturally singularity free theory for isolated bodies (as was cherished by Einstein). This M->0 state would materialize after the body radiates its entire initial mass-energy. Thus there is no event horizon at any finite R or M, and, therefore all the great theoretical confusions like whether there could be (i) White Holes, (ii) whether t ® becomes spacelike (timelike) inside the EH (iii) Loss of information in gravitational collapse, and (iv) validity of cosmic censorship conjecture get resolved. At any finite proper time, the collapsed object would be either static (z<2) or may appear static (R almost frozen) though, in extreme cases, internally, in terms of proper radial length, it might be collapsing at a speed ~c! We call the latter as Eternally Collapsing Objects.

 

later 2006 he writes this paper.

 

 

Sources of Stellar Energy, Einstein- Eddington Timescale of Gravitational Contraction and Eternally Collapsing Objects

[astro-ph/0608178] Sources of Stellar Energy, Einstein- Eddington Timescale of Gravitational Contraction and Eternally Collapsing Objects

 

 

Authors: Abhas Mitra

 

(Submitted on 8 Aug 2006 (v1), last revised 3 Sep 2006 (this version, v3))

Abstract: We point out that although conventional stars are primarily fed by burning of nuclear fuel at their cores, in a strict sense, the process of release of stored gravitational energy, known as, Kelvin - Helmholtz (KH) process is either also operational albeit at an arbitrary slow rate, or lying in wait to take over at the disruption of the nuclear channel. In fact, the latter mode of energy release is the true feature of any self-gravity bound object including stars. We also highligh the almost forgotten fact that Eddington was the first physicist to introduce Special Relativity into the problem and correctly insist that, actually, total energy stored in a star is not the mere Newtonian energy but the total mass energy (E = M c^2). Accordingly, Eddington defined an ``Einstein Time Scale'' of Evolution where the maximum age of the Sun turned out to be t_E = 1.4. 10^{13} yr. We extend this concept by introducing General Relativity and show that the minimum value of depletion of total mass-energy is t_E =infty not only for Sun but for and sufficiently massive or dense object. We propose that this time scale be known in the name of ``Einstein - Eddington''. We also point out that, recently, it has been shown that as massive stars undergo continued collapse to become a Black Hole, first they become extremely relativistic Radiation Pressure Supported Stars. And the life time of such relativistic radiation pressure supported compact stars is indeed dictated by this Einstein -Eddington time scale whose concept is formally developed here. Since this observed time scale of this radiation pressure supported quasistatic state turns out to be infinite, such objects are called Magnetospheric Eternally Collapsing Objects (MECO).

 

My question to this is what is the state or phase of matter in these star cores to be able to store gravitational radiation?

[Jeffocal]

Dear Modest and Pluto

 

I believe Einstein was vehemently opposed to the whole concept of a black hole. I am pretty sure his opposition was was not just based on a dislike but on an inconsistency between its existence and his theoretical concepts. Unfortunately I don't think we, being mere mortals, are smart enough to figure out why.

 

We know there is an problem with respect to time in that someone who is watching a mass implode views the ttime dilation on it surface to be infinite when it passes the event horizon. However even though time has stopped for an external observer many think that it can continue to collapse. I know there are logic and consistent arguments that can explain it in terms of relativistic concepts but I believe they are not based on observational reality. If you don't minid I'd to spend some time thinking about what your taught me and then continue this subject.

 

In the mean time could you please try to explain to me why Einstein chose to define the universe in terms of four dimensional space-time instead of four spatial dimensions.

 

For the past couple of years I have been tring to transpose his space-time geometry to four *spatial* dimensions. I've compiled the results at The Imagineer’s Chronicles

 

I'd be interested in a second opinion

 

Thanks friends

[Jeffocal]

Dear Modest and Pluto

 

In the mean time could you please try to explain to me why Einstein chose to define the universe in terms of four dimensional space-time instead of four spatial dimensions.

 

For the past couple of years I have been tring to transpose his space-time geometry to four *spatial* dimensions. I've compiled the results at The Imagineer’s Chronicles

 

I'd be interested in a second opinion

 

Thanks friends

 

I will, if anyone feels that it would be more approperate generate a new thead to disscuss this topic in another catogory. However, it may be benifical to compare then side by side with the subject material of this thread becuase they have their foundation in Einstien's brilliant descirption of a space-time geometry.

 

Jeff

[Jeffocal]

Modest

 

Is there anything in Einstein's relativistic theories that would prevent them from being applied to the gravitational forces of a star as an isolated system.

 

Jeff

[modest]

I believe Einstein was vehemently opposed to the whole concept of a black hole.

 

Yes. I've read that he originally ignored them, then moved to actively arguing against their physical existence.

 

I am pretty sure his opposition was was not just based on a dislike but on an inconsistency between its existence and his theoretical concepts.

 

I don't think so. His argument, as best as I can recall, was that the angular momentum of some collapsing system would stabilize it before a black hole could form. The physics of black holes is sound in so far as General Relativity is concerned. The real question was whether or not conditions could exist in nature that would form them.

 

There is quite a bit of astronomical evidence that indicates they do indeed form.

 

In the mean time could you please try to explain to me why Einstein chose to define the universe in terms of four dimensional space-time instead of four spatial dimensions.

 

I will, if anyone feels that it would be more approperate generate a new thead to disscuss this topic in another catogory. However, it may be benifical to compare then side by side with the subject material of this thread becuase they have their foundation in Einstien's brilliant descirption of a space-time geometry.

 

That is a rather large topic. If you'd like to start a new thread that would be fine, but I don't mind discussing it here if you think it's related.

 

There are normally only considered to be 3 spatial dimensions simply because there don't appear to be more kinematically. Looking at things macroscopically, they appear to have height, width, and depth and no other spatial directions.

 

Time used to be considered something like an evolution paramater attached to every point in space (or every object in space). But, shortly after Einstein developed general relativity Minkowski figured out a neat geometric way of representing time as a 4th dimension making what we now call spacetime. I should add: I apologize if you already know all this—I don't mean to sound patronizing.

 

In Minkowski spacetime (and later in GR) time is attached to space geometrically, but it is slightly different from the other 'spatial' dimensions. The first important distinction is what is represented in the geometry. In ordinary 3D space we think of locating objects. We might say, for example, that Alpha Centari is about one parsec from the sun. A 3D spatial geometry will tell you how far apart the two objects (the two stars) are from one another.

 

In spacetime it's not objects which are located on the geometry but "events". An event is a point in space and time. An example would be Neil Armstrong taking his first step on the moon. So, the very concept of what the metric measures is different in spacetime versus space. Spacetime can give you the space-time distance between events. If a rocket takes off from earth and lands on the moon then it has traveled through space and time between the two events (from our perspective).

 

In order for this system to work usefully we have to mathematically treat space differently from time. To find distance through space we use the Pythagorean theorem. Where we have an x, y, and z axis we can calculate the distance between two points like so:

 

[math]\mbox{distance} = \sqrt{x^2 + y^2 + z^2}[/math]

 

Notice we added the x, y, and z parts together. The geometry must change a little when we add the forth (time) dimension. Distance becomes:

 

[math]\mbox{distance} = \sqrt{x^2 + y^2 + z^2 - t^2}[/math]

 

Time (t) was subtracted rather than being added like the spatial dimensions. This may seem like a small matter, but in terms of geometry it is significant. It means that time is not exactly like the spatial dimensions. It does not behave in exactly the same way. I find this naturally appealing. Time does not, after all, seem intuitively to act exactly like the spatial dimensions, so it only seems natural that our physics treat time a little differently.

 

I don't know what General Relativity with 4 space instead of 3+1 spacetime would look like or how it really would even make sense. It gets down the the fundamentals of what an event is: something with a location in space and a location in time. With 4 spatial dimensions I don't know what GR would be describing: 4 dimensional objects?

 

Modest

 

Is there anything in Einstein's relativistic theories that would prevent them from being applied to the gravitational forces of a star as an isolated system.

 

I don't know what you mean by "isolated system", but Schwarzschild's solution to GR describes the gravitational field outside a non-rotating spherical mass which works very well for considering a planet or star. It's also a vacuum solution which I suppose would make it an "isolated" system.

 

~modest

[Pluto]

G'day from the land of ozzz

 

This paper is related to your discussion. The ABS says what I want to say.

 

[0905.2575] Motion in alternative theories of gravity

Motion in alternative theories of gravity

 

Authors: Gilles Esposito-Farese

(Submitted on 15 May 2009)

 

Abstract: Although general relativity (GR) passes all present experimental tests with flying colors, it remains important to study alternative theories of gravity for several theoretical and phenomenological reasons that we recall in these lecture notes. The various possible ways of modifying GR are presented, and we notably show that the motion of massive bodies may be changed even if one assumes that matter is minimally coupled to the metric as in GR. This is illustrated with the particular case of scalar-tensor theories of gravity, whose Fokker action is discussed, and we also mention the consequences of the no-hair theorem on the motion of black holes. The finite size of the bodies modifies their motion with respect to pointlike particles, and we give a simple argument showing that the corresponding effects are generically much larger in alternative theories than in GR. We also discuss possible modifications of Newtonian dynamics (MOND) at large distances, which have been proposed to avoid the dark matter hypothesis. We underline that all the previous classes of alternatives to GR may a priori be used to predict such a phenomenology, but that they generically involve several theoretical and experimental difficulties.

[freeztar]

This paper is related to your discussion. The ABS says what I want to say.

 

Pluto, can you please explain the relevancy of the link you posted to the current discussion?

[Jeffocal]

Yes. I've read that he originally ignored them, then moved to actively arguing against their physical existence.

 

I don't think so. His argument, as best as I can recall, was that the angular momentum of some collapsing system would stabilize it before a black hole could form. ~modest

 

I believe he could have used a different argument based on the fact that relativity defines the magnitude of a space-time curvature in a volume of space as being dependent on the magnitude of the gravitational potential in that volume.

 

However, the gravitational potential at the center of a mass is by definition zero therefore the according the concepts of relativity the space-time volume in the center of a collapsing star must remain flat. Additionally the space-time curvature caused by a gravitational potential would gradually increase as one moved outward from its center because the gravitational potential does. The fact that the space-time curvature at the center of a collapsing star must according to relativity remain flat suggests that space must contain some form of energy which counteracts the curving effects of a gravitational potential because the concepts of relativity mandates as just shown there is a limit to its ability to generate a space-time curvature.

 

Even though Einstein could not define its origins and how it interacted with a gravitational potential he knew it had to exist.

 

One reason why he couldn't may have been because it was not related to time or space-time property of space but to a spatial property of four *spatial* dimension.

 

Below is a reprint of the article Dark energy: the cosmological constant which appeared in the Imagineer's Chronicles. I believe it defines a mechanism in terms of four spatial dimension which may be related to the one Einstein was looking for.

 

"We have shown throughout this blog observations of our environment indicate the universe is composed of four *spatial* dimensions rather than of four-dimensional space-time as is suggested by Einstein's theories.

 

The recent discovery of Dark energy or a cosmological constant is one of those observations.

 

As Alexey Vikhlinin of the Smithsonian Astrophysical Observatory in Cambridge, Mass. wrote in the Scientific Frontline article Dark Energy Found Stifling Growth in Universe Dec 16, 2008:

 

This "study strengthens the evidence that dark energy is the cosmological constant. Although it is the leading candidate to explain dark energy, theoretical work suggests it should be about 10 raised to the power of 120 times larger than observed. Therefore, alternatives to general relativity, such as theories involving hidden dimensions are being explored.

 

Putting all of this data together gives us the strongest evidence yet that dark energy is the cosmological constant, or in other words, that 'nothing weighs something'," said Vikhlinin. "A lot more testing is needed, but so far Einstein's theory is looking as good as ever."

 

But the fact that Dark Energy or a Cosmological constant was discovered does not alter the fact Einstein introduced his cosmological constant because he was trying to make or "force" the universe conform to his expectations, rather than using theory to guide him to an understanding of its properties.

 

In the article "Why Space time?" Sept. 27, 2007 is was shown that it is possible to explain and predict all of the observed properties of relative motion including time dilatation, length contraction and mass increase as logically and consistently in terms of the geometry of four *spatial* dimensions as can be done in terms of the four-dimensional space-time geometry of Relativity.

 

Additionally it was show in the article “Gravity” Dec 15, 2007 it is possible to consistently define all the observed properties of a gravitational field in terms of a curvature in "surface" of a three-dimensional space manifold with respect to a fourth *spatial *dimension.

 

However, theoretically defining the universe in terms of four *spatial* dimensions as we have done in this blog and the paper The "Shadows" of four *spatial* dimensions means one does not have to "force" the integration of Dark energy or a cosmological constant into its theoretical structure as Einstein had to do to integrate it into Relativity.

 

If a gravitational field is a result of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension, as we are suggesting then reducing the mass of an object would reduce the magnitude of that curvature. However, this would result in the expansion of the "surface" of that three-dimensional space manifold. This would result in a physical expansion of the volume of the universe analogous to how removing the coils or curvature in a rope causes a physical expansion of its length with respect to three-dimensional space.

 

However, observations of our universe indicate Dark Energy expands the "surface" of space in a similar manner. This means that one can use a theory based on the existence of four *spatial* dimensions to guide them to an understanding of its properties rather than just expectations as Einstein had to.

 

Since the total mass in the universe is decreasing due to the nuclear reactions taking place in stars, the curvature associated with its gravitational field would also be decreasing. This means, according to this theoretical model the magnitude of this component of dark energy would be proportional to the quantity of mass the universe losses due to the nuclear reactions that occur within it.

However, the "concentration" of Dark Energy relative to gravitational energy would be, according to this mechanism defined by the equation E=mc^2c. This means its strength should be 1/c^2 weaker than gravitational forces. Therefore, because gravitational forces are much stronger than those of dark energy or the cosmology constant, the space between gravitationally bound objects would not appear to be expand.

 

This is consistent with the observation that its magnitude is about 10 raised to the power of 120 times smaller than predicted by Einstein's theories" and the fact that galaxies are not observed to be expanding, just the space between them. "

 

Later Jeff

 

Copyright 2009 Jeffrey O'Callaghan

[Jeffocal]

Dear Modest

 

I'd like to thank you for taking the time to help me understand the depth of Einstein's genius.

 

I would also like to ask you a favor. I am receiving a military disability for a head trauma that affects my ability to comprehend explanations presented in a mathematical format. Could please, wherever possible try to use words to explain concepts.

 

Einstein quantified the relativistic properties of space, time and gravity in terms of a curvature or displacement in a space-time manifold, in which time and space varies in accordance with the magnitude of that displacement.

 

I have been attempting to use qualitative arguments to show how it would be possible to predict those relativistic properties in terms of a spatial displacement in a "surface" of a three dimensional space manifold with respect to a four *spatial* dimension.

 

The displacements responsible for the relativistic properties of space and time in both environments have a common element in the three spatial dimensions; therefore, their performance will be identical. The only difference between them is that one defines a displacement in terms of a time variable while the other in terms a variable spatial distance.

 

However, Einstein based the relativistic properties of time on the fact that the speed of light is constant for all observers. But distance equals velocity times time. Therefore could one quantitatively define the spatial distance a three dimensional space manifold moves with respect to a fourth *spatial* dimension by multiplying the speed of light with the relativistic time change predict by Einstein for a given situation.

 

If so could one plug that value into the field equation that Einstein developed to quantify his concepts in terms of existence of four *spatial* dimension instead of four-dimensional space time.

 

Thanks again Jeff

[modest]

However, the gravitational potential at the center of a mass is by definition zero therefore the according the concepts of relativity the space-time volume in the center of a collapsing star must remain flat.

 

Not exactly. The value of potential does not determine the shape of the field, but rather how quickly potential changes. In a plot of potential in a star:

 

 

where the blue line is pretty much horizontal, space is pretty well flat. That's at the far left and the far right of the blue line (the center of the star and infinitely far from the star). These two areas have very different values of potential, but both have flat space because the shape (or strength) of the field is not determined by the value of potential, but how quickly potential changes with distance. The force of gravity is the derivative of gravitational potential energy.

 

Additionally the space-time curvature caused by a gravitational potential would gradually increase as one moved outward from its center because the gravitational potential does. The fact that the space-time curvature at the center of a collapsing star must according to relativity remain flat suggests that space must contain some form of energy which counteracts the curving effects of a gravitational potential because the concepts of relativity mandates as just shown there is a limit to its ability to generate a space-time curvature.

 

Again, you're working under the assumption that the *value* of potential determines the force of gravity (or the strength of the field). That is not the case. You might think: "how much does gravitational potential change from one spot in space to the next spot in space?" If the change is not very great then the field is weak and nearly flat. If the change is a lot then the field is strong.

 

If a gravitational field is a result of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension...

 

This is fine. Relativity describes 3 curved dimensions. If you want to say that they curve into a 4th dimension that is somehow hidden to us then that's not an unusual thing to think. It would be called extrinsic curvature. But, you still need time as a dimension—a curved dimension. It is the curvature of time that results in the gravity we feel here on earth. It is (as far as I know) an indispensable part of general relativity.

 

Dear Modest

 

I'd like to thank you for taking the time to help me understand the depth of Einstein's genius.

 

I would also like to ask you a favor. I am receiving a military disability for a head trauma that affects my ability to comprehend explanations presented in a mathematical format. Could please, wherever possible try to use words to explain concepts.

 

Sure.

 

The displacements responsible for the relativistic properties of space and time in both environments have a common element in the three spatial dimensions; therefore, their performance will be identical. The only difference between them is that one defines a displacement in terms of a time variable while the other in terms a variable spatial distance.

 

Well... if you're saying that space (the 3 normal dimensions that we know) is curved into a 4th 'hidden' dimension rather than being curved into time (which is the usual interpretation) then I think that's ok. You could also say that time has extrinsic curvature into this higher dimension.

 

However, Einstein based the relativistic properties of time on the fact that the speed of light is constant for all observers. But distance equals velocity times time. Therefore could one quantitatively define the spatial distance a three dimensional space manifold moves with respect to a fourth *spatial* dimension by multiplying the speed of light with the relativistic time change predict by Einstein for a given situation.

 

If so could one plug that value into the field equation that Einstein developed to quantify his concepts in terms of existence of four *spatial* dimension instead of four-dimensional space time.

 

Thanks again Jeff

 

I don't really follow what you're saying there. The speed of light works as a conversion factor between space and time. To switch from space units to time units (or from time to space) you multiply or divide by the speed of light. There is no conversion factor between spatial dimensions, so this once again shows that time is a little different from the other dimensions.

 

~modest

[Jeffocal]

This is fine. Relativity describes 3 curved dimensions. If you want to say that they curve into a 4th dimension that is somehow hidden to us then that's not an unusual thing to think. It would be called extrinsic curvature. But, you still need time as a dimension—a curved dimension. It is the curvature of time that results in the gravity we feel here on earth. It is (as far as I know) an indispensable part of general relativity.

 

~modest

Would it still be necessary to define time as a dimensions if one defined it only in terms of a measure of the sequential ordering of the casualty of events and then defined the relativistic curvature Einstein associated with its dimensional properties in terms of a curvature in four *spatial* dimensions as we have tried to do in the following article.

 

BTW you can find a complete listing of 58 different subjects we have discussed to point at Articles posted after 01-09 | The Imagineer's Chronicles

 

Thanks

Jeff

 

The Relativity of four *spatial* dimensions

 

We have shown in this blog that there is more observational evidence supporting the existence of four *spatial* dimensions than four-dimensional space-time.

 

One of these observations is the relativistic properties of space and time.

 

The article "Defining energy" Nov. 26, 2007 showed one could define a mechanism responsible for gravitational and kinetic energy that is more consistent with observations in terms of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension than one in four-dimensional space-time manifold.

 

However, this also gives one the ability to define a mechanism responsible relativistic length foreshortening associated with a gravitational field and velocities in terms of a curvature in a "surface" of a three-dimensional space manifold.

 

This is because, as was shown in the article "Embedded Dimensions" Oct. 22, 2007 three-dimensional beings would only able to observe the cord of the arc generated by a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

 

Therefore, the length of objects will appear foreshortened when viewed from a gravitational field because the cord of the arc generated by a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension by a gravitational field will be shorter than the arc itself.

 

The same would be true for objects in relative motion because as was shown in the article "What is energy" Nov. 26, 2007 the kinetic energy of their motion is a result of an "oppositely directed" curvature in a "surface" of a three-dimension space manifold with respect to a fourth *spatial* dimension. Therefore, each observer would view the length of an object in relative motion as being shorter because they will view the cord of the arc generated by a curvature in the "surface" of a three-dimensional space manifold caused by the relative motion of the objects.

 

However, the fact that three-dimensional beings can only view the cord of the arc in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by kinetic or gravitational energy will also affect the movement of time in all reference frames.

 

In the article "Defining time?" Sept 20, 2007 it was shown defining it only in terms of a measure of the sequential ordering of the causality of an event would provide an unambiguous definition of time that is consistent with both the physical and mathematical observations of time.

 

If this is a valid definition, time would be dilated with respect to an external reference frame that was in motion or in a gravitational field because of the foreshortening of length, the measurement of the distance between events would be greater for an observer in that reference frame than for one who is outside of it. Therefore, time in a gravitatonal or moving reference frame would dilate or move slower with respect to one outside of it because the ordering of the causality of those events in those reference frames would take longer to complete than they would with respect to an external observer becuase they are seperated by a greater distance.

 

This indicates one can explain and predict the relativistic properties of gravity, space and time in terms of a curvature in a "surface" of a three-dimensional space manifold as logically consistently as can be done in terms of a curvature in four-dimensional space-time manifold.

 

When you combine these observations regarding time dilation, gravitational and kinetic energy with the others presented in this blog it becomes extremely difficult for anyone to dispute that fact the universe is composed of four *spatial* dimensions instead of four dimensional space-time.

 

Later Jeff

 

Copyright 2007 Jeffrey O'Callaghan

[Jeffocal]

I don't really follow what you're saying there. The speed of light works as a conversion factor between space and time. To switch from space units to time units (or from time to space) you multiply or divide by the speed of light. There is no conversion factor between spatial dimensions, so this once again shows that time is a little different from the other dimensions.

 

~modest

 

I guess what I am tiring to say is that Einstein defined (I hope I get this right :confused: ) the magnitude of a gravitational potential in terms of a curvature in a surface of a space-time manifold and what we are saying is that it is a result of a curvature in a "surface" of a three-dimensional manifold with respect to a fourth *spatial* dimension. I guess I am asking is there a way of mathematically converting the magnitude of a space time curvature to an equivalent spatial distance. If so could that be used to define the magnitude of the distance a "surface" of a three-dimensional space manifold would be displaced with respect to a fourth *spatial* dimension to cuase an equivalent space-time curviature.

 

Then could we substitute that value in for the values that represent the space-time curvature in Einstein's field equations to quantify them in terms of four *spatial* dimensions.

 

Jeff

 

PS Would it be possible to use the fact that magnitude of time dilation and length foreshortening are connected in terms of the same space-time curvature to quantify an equivalent one in a four dimensional manifold???

[modest]

I guess what I am tiring to say is that Einstein defined (I hope I get this right :shrug: ) the magnitude of a gravitational potential in terms of a curvature in a surface of a space-time manifold and what we are saying is that it is a result of a curvature in a "surface" of a three-dimensional manifold with respect to a fourth *spatial* dimension. I guess I am asking is there a way of mathematically converting the magnitude of a space time curvature to an equivalent spatial distance. If so could that be used to define the magnitude of the distance a "surface" of a three-dimensional space manifold would be displaced with respect to a fourth *spatial* dimension to cuase an equivalent space-time curviature.

 

Then could we substitute that value in for the values that represent the space-time curvature in Einstein's field equations to quantify them in terms of four *spatial* dimensions.

 

Sorry for the delayed response.

 

I still don't know what you're looking for exactly. According to general relativity, 3 dimensions of space and one dimension of time are curved. They are all 4 curved. They theory does not say what they are curved into. If you want to say that they are curved into a higher dimension then that's fine. It would not change the predictions or the structure of the theory at all. It's the difference between extrinsic and intrinsic curvature. Extrinsic curvature is the curvature of a surface into a higher dimension. Intrinsic curvature is a curved surface (regardless of higher dimensions). It makes no difference if the intrinsically curved spacetime in general relativity has a 5th dimension which it is extrinsically curved into or not. Notice:

It is important to realize that the local geometry or curvature characterized by (2.4) [Einstein’s field equation] is an intrinsic property of the manifold itself, i.e. it is independent of whether the manifold is embedded in some higher-dimensional space.

 

General relativity: an introduction ... - Google Books

 

The important thing is that space and time are both curved. Whatever else you want to say about what they are curved into is fine, but should not change the particular answers you get with the theory.

 

~modest

[Jeffocal]

Maybe I should have used the term distorted instead of curved because and I think Einstein would agree that would be more accurate way of describing how both the geometry of space-time and four *spatial* dimensions is altered by mass.

 

Einstein derived equation that quantified the magnitude of length contraction and time dilation based on the total energy content of an object. The greater the velocity of an object the greater its energy and therefore the greater the relative time dilation and length shortening that object will experience due to the energy of its velocity.

 

However, correct me if I am wrong but this also means that Einstein defined the causality of all mass and energy in terms of a geometric distortion of both a spatial or length and time component of a space-time manifold.

 

What I am trying to says is does the fact that he analytically defined the effects energy has on space or length and time in terms of a common mechanism related to energy provide us with an connection between his space-time geometry and one of four *spatial* dimensions.

 

Would this provide us with a way of analytically determining if it would be possible to transpose his space-time geometry to four *spatial* dimensions without altering any of its predictive ability.

 

Jeff

[modest]

I'm sorry Jeffocal. Despite giving it my best, I just can't make sense of what you are saying.

 

Perhaps someone else has insight....

 

~modest