So in the diagram, m is a point mass sitting in 1 dimensional space (labeled Schwarzschild space)? Those other curves (local time, local space), should we project them onto the schwarzchild line space to get the respective GR curvature (which I assume would be a tensor (since it would have more than one value at a single point it can't be a scalar)?

In 4 dimensional space-time the Schwarzschild metric has at least 10 space-time curvatures. How many space-time curvatures would be present for this 2 dimensional space-time?

Hi Roger, you asked: "Those other curves (local time, local space), should we project them onto the schwarzchild line space to get the respective GR curvature ... "

I don't think this projection is going to give curvatures - it would merely give us the coordinates of any point in Schwarzschild spacetime, which would be 4 signed values in 3+1 spacetime.

The issue of 10 'curvatures' in the Schwarzschild metric is a bit 'foggy' - there are actually 12 non-zero Christoffel symbols that make up the metric (http://www.mathpages.com/rr/s5-05/5-05.htm), but the last four "reversibles" are normally counted as just two . And because "For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives" it is sometimes stated as 10 curvatures, or sometimes as 10 potentials. :(

In the static view of the metric (as in my diagram), only the first four of the Christoffel symbols shown in http://www.mathpages.com/rr/s5-05/5-05.htm are applicable - the ones with only t and/or r subscripts, because there cannot be transverse (tangential) movement or accelerations. The other six comes into play when dphi and dtheta are non-zero. Looked at simplistically, there are really only two curvatures for any point in my diagram (time curvature and space curvature), until that is, you let a particle free-fall from that point - then there are more...

The wonderful generality of Einstein's theory sometimes makes it very 'foggy'.

-J

Ok, sorry, I was using sloppy language with "project". Lets just avoid that statement I made because I'm starting to capture a glimmer of the fuzziness you're talking about.

In the diagram you provided, the space curve and time curve look the same, just one is on the x-axis and one is on the y-axis. Does that change when velocity of a test particle is involved?

Also, you said the first four Christoffel symbols, you mean the ones indexed tr, rt, tt, and rr, right? Because in a spacetime line there is no theta or phi, right. I think I'm starting to get the basics.

Hi Roger,

No, the time- and space-curves have different equations. The time curve is "time curvature" and the space curve is "spatial curvature". Both satisfies the Schwarzschild metric for static particles and are given respectively by: time τ = √(1-r_s/r) (gravitational time dilation) and space: z = 2√[r_s(r-r_s)] (Flamm's paraboloid). Obviously not the same thing.

When the velocity of a test particle is involved, it is not so much the curves that change, but rather the position of the curvature center, but that's quite a complicated issue, unfortunately outside the scope of the present thread. This purpose here is to simply make clear what spacetime curvature is in its simplest form.

-J

Ok, I see going to have to understand Christoffel symbols if I ever want to understand this. So...

In this link it explains the nonzero Christoffel symbols you are talking about:

http://www.mathpages.com/rr/s5-05/5-05.htm

However the notation on that page seems to be of Christoffel symbols of the second kind:

yet the answers seem to be that of Christoffel symbols of the first kind:

http://en.wikipedia.org/wiki/Christoffel_symbols#Relationship_to_index-free_notation

So I guess what I'm asking is, which are used in general relativity, Christoffel symbols of the first kind, second kind, or both depending upon the situation.

Hi Roger,

I do not see any notation of Christoffel symbols of the first kind in the mathpages link. (?)

AFAIK, GR only uses the second kind, the "connection coefficients" as MTW (Misner et al. 1973, p. 210) calls them.

-J

You answered my question. So GR uses only Christoffel symbols of the second kind.

So for the 4 dimensional spacetime manifold there are 64 possible Christoffel symbols (counting zero and non-zero)?

What is the Weak Field Approximation? I think I'm seeing that it is used in the derivation of the Schwarzschild metric but I don't understand exactly what it is. Is it used to get the Schwarzschild metric?

Hi Roger,

The 'weak field limit' is usually taken as Newtonian gravity, but 'weak field approximation' may mean simplified GR in that second order terms in a series expansion are ignored, e.g. √(1-2GM/rc²) ≈ 1-GM/rc² for rc² « GM, getting rid of the square root.

AFAIK, the mathpages derivation uses the 'weak field limit' only to find some constants of integration, because GR must approach Newton gravity at large ranges and gravity must approach zero as r goes toward infinity.

The biggest leap of faith is taken by assuming that a circular orbit in Schwarzschild geometry is effectively the same as in Newton's gravity (in terms of orbital period for a given radius). This is one of those fortuitous cases where gravitational time dilation and curvature effects cancel each other. More rigorous derivations of the Schwarzschild metric (none of which I quite follow) apparently proves that this is the case. I have my own 'engineering way' of demonstrating this in Relativity-4-Engineers, chapter 5, (second link is directly to the chapter's pdf) page 82, but I guess one would need to read the prior part of the chapter to make sense out of it.

One must also remember, that once a solution to the field equations is obtained, taking whatever leaps of faith one desires, one can substitute it back into the field equations and see if it actually solves them. Apparently it does - I take the mathematician's word for that.

-J

Oops, noticed it too late to edit the 'slip of the click' in #10. Should obviously be:

e.g. √(1-2GM/rc²) ≈ 1-GM/rc² for rc² » GM...

J

Hi Jorrie,

I find your drawing intimidating even without the math. Is Schwarzchild space the line between the two planes? What is the difference between local space and Schwarzchild space? I could only find Schwarzchild space-time when I Googled.

After just watching the movie IQ with Walter Matthau playing Einstein, I will pretend I am putting a pipe in my mouth. I say "That is an interesting concept" and "goodnight"

-S

Hi S,

Yes, it is the line where the two planes meat, representing normal 3-D space, but perhaps not the best choice of terminology. More correct would have been something like "spatial portion of Schwarzschild coordinates"; or: "space as observed from infinity"; or: asymptotically flat space". None very satisfactory, I must confess. For all I know, there may be a totally different beast that is officially called "Schwarzschild space".

Anyway, the radial parameter r of the standard Schwarzschild metric is a projection of the curved space, which I called "local space" onto flat space, which I called "Schwarzschild space". In this sense, "Schwarzschild space" has no existence, it is just a mathematical convenience; only "local space" exists - it is where we live in.

I find 'curved space' less confusing than 'rubber measuring rods' (which contract as they get closer to a gravitating body), but that's an equally valid view of what happens, because it is only the observation that is 'real', whatever that may mean...

-J

Hi Jorrie,

It is my understanding that gravitational lensing effect can be demonstrated from the schwarzchild solution (4 dimensional case). Although gravitational lensing would be impossible for 2 dimensional space time, the cause of the effect in 4 dimensional spacetime is present (I think) in the description of the radial part of the 2 dimensional metric:

dτau² =(1-r_s/r) dt² - dr²/(1-r_s/r)

Can you explain this effect for us to show how it works?

Roger

Hi Roger,

I did write a little about gravitational lensing in a Blog post (or two) way back in 2007, so I guess you might first want to check them out :)

In a few words, the 'bending' of light is performed on an equal (50:50) basis by the first two terms of the Schwarzschild metric, time curvature and spatial curvature respectively. The first half is sometimes described as "due to the equivalence principle", because Newton's gravity also predicts that half, provided one uses the corpuscular theory of light (essentially photons), not the wave theory.

As a quantum theorist, you may be interested in this view (haven't seen it published anywhere, so pure fun on my part :)

If we consider space to be denser nearer to a massive body, then the wave theory could provide "the other half" of the bending for Newton. So, because of the wave/particle duality of photons, maybe it is natural that they do both: a photon as particle is bent by the equivalence principle (time curvature) and the photon as wave is bent by spatial curvature, so viola, we have double bending of light.

-J

Interesting stuff, I will read your post.

You wrote:"In a few words, the 'bending' of light is performed on an equal (50:50) basis by the first two terms of the Schwarzschild metric, time curvature and spatial curvature respectively."

But I think you meant "time curvature and radial curvature", is that correct?

You Wrote:"If we consider space to be denser nearer to a massive body, then the wave theory could provide "the other half" of the bending for Newton."

Interesting. It would be like a gradient index lens. Some manufacturers will vary the density of a glass to vary the index or refraction which produces a lensing effect:

The green curve on the left is the index of refraction (largest in the middle). The darker shading of blue in the center (and lighter shades towards the edges) represent the varying density of the material.

You Wrote:"because Newton's gravity also predicts that half, provided one uses the corpuscular theory of light"

I'm trying to understand, so Newton's gravity predicts time curving for the particle interpretation of light. Is this because light isn't massless in the corpuscular theory of light and thus experiences a gravitational pull (slows down?)

Hi Roger: "But I think you meant "time curvature and radial curvature", is that correct?"

I suppose one can call it "radial spatial curvature" if you like.

"I'm trying to understand, so Newton's gravity predicts time curving for the particle interpretation of light. Is this because light isn't massless in the corpuscular theory of light and thus experiences a gravitational pull (slows down?)"

Maybe "time curving" was putting it a bit strongly, because Newton's gravity surely does not affect time. It simply does the same job as Einstein's time curvature (first 50%). Lastly, yes, Newton will have to assign an infinitesimal mass to the corpuscular photon and let it travel at very near c. Then it will follow an open (hyperbolic) orbit around the Sun. He just could not have predicted the other 50%, unless he also speculated on "denser space" around the Sun.

-J

If we consider space to be denser nearer to a massive body, then the wave theory could provide "the other half" of the bending for Newton. So, because of the wave/particle duality of photons, maybe it is natural that they do both: a photon as particle is bent by the equivalence principle (time curvature) and the photon as wave is bent by spatial curvature, so viola, we have double bending of light

That is an interesting concept (no pretend pipe now). The idea (which I made bold) has credibility in view of the Casimir Effect. But what could be causing the space to be denser near a massive object other than "Newton's gravity"?

Hi S: "But what could be causing the space to be denser near a massive object other than "Newton's gravity"?"

Einstein's gravity, I suppose. But then Newton did not know about it, so he would have had to pull such a theory out of the bag. It is interesting to think about what Newton would have done if some astronomer did measure the deflection of light by the Sun in his time. :))

-J

I have a naive question, why is the metric expressed in terms of proper time? I've seen it can be expressed in terms of ds² like this:

and that is how I would expect it to be given, yet wikipedia gives the metric as you do, in terms of proper time:

In general, when things like this are done it's for good reason, so what is gained by expressing the metric in terms of proper time?

Hi Roger,

You obviously know that spacetime interval (squared) ds² = -(cdτ)², so the question is, why is it convenient to write the metric in the cdτ form?

The only answer I have is that some like it that way, me included. My main motive is that it makes the relationship between proper time (τ) and time (t) very easy to spot. Divide the both sides by (c dt)², massage a little bit and you have: (dτ/dt)² = (1 - r_s/r)(1 - (v/c)²), where v is a locally measured (3-D) velocity vector. So, the Schwarzschild time dilation can be viewed as a gravitational time dilation multiplied by a velocity time dilation. If this is not too clear, I suggest that you reread chapter 1 of Relativity-4-Engineers, especially from section 1.3 (pp. 23 to 29), because it is too bulky to put in here.

Most engineers have a slight fright when they first read that chapter,⁽¹⁾ but with your background, I'm sure that you found it quite easy. With hindsight, I should have relegated it to an appendix and should have replaced it with something more palatable, but I regrettably did not... For those who do not have the eBook, here is a link to chapter 1 (pdf).

-J

From your eBook (chapter 1):

"Some authors prefer to work solely with the timelike interval as the metric of spacetime (e.g. Faber). This is presumably because most of the intervals that we can observe and measure are timelike."

That makes sense to me, thanks.

For the 2 dimensional space-time example provided in this post, given

I believe in this case Rμν and gμν are two by two tensors. The later being a diagonal tensor where where grr=(1-r_s/r)^-1 and gtt=(1-r_s/r), is that correct?

If so, what would be components of Rμν and what would be the value of the scalar R. Is R (the scalar) what you calculated in note #1?

Roger asked: "I believe in this case Rμν and gμν are two by two tensors. The later being a diagonal tensor where where grr=(1-r_s/r)^-1 and gtt=(1-r_s/r), is that correct?"

I think so, yes.

I think the components of Rμν would be the first four Christoffel symbols in that Mathpages link we discussed.

The R that I used in footnote (1) of the main post is just a geometric radius of geodesic curvature (vector) and not the Ricci scalar (which is zero, because Schwarzschild is a vacuum solution, as you know). My R's definition is more or less given in main post: "Now the interesting thing is that if we let time move on (by an increment dt) and let the vector R rotate around the curvature center cc, the centripetal acceleration at the point r_o is exactly equal to the proper gravitational acceleration at that point."

I should probably have used a different symbol, avoiding confusion with the Ricci scalar.

-J

Thanks for the blog Jorrie. As usual, very humbling. I'm glad we've got guys like you working on these things.

Pictures always help.

Now the interesting thing is that if we let time move on (by an increment dt) and let the vector R rotate around the curvature center cc, the centripetal acceleration at the point r_ois exactly equal to the proper gravitational acceleration at that point.
Does this mean that the passage of time is responsible for gravitational acceleration (i.e. gravity itself)? Or possibly vice-versa, or one can't exist without the other?
I'm trying to wrap my mind around curvature center. Since the universe doesn't have a center, are you talking about space in the presence of a massive object?

Hi S, I'm out of town with only B/Berry internet, so I will do fuller reply later. One can say that time curvature is at least partly causing gravity. Add spatial curvature and you have all. :-) -J

Hi S, I've got guest house WiFi and laptop talking to each other now; much better than BB internet. :)

What we perceive as gravity can be explained as curved spacetime, but other explanations would also do the trick, e.g. denser spacetime or perhaps gravitons conveying a force. What we do know is that when we observe the effects of gravity, we also observe the slowing down of clocks and the delay of signals, effects best explained by curved spacetime.

The spacetime curvature center discussed here is of the local variety, where it is unique for each distance from a central mass (and for each velocity relative to the mass). The universe as a whole has only a (hypothetical) average center of curvature, based on the average spacetime curvature. We know the curvature is very close to zero and hence near-infinite or perhaps even an imaginary (complex) radius of curvature. We simply cannot measure accurately enough yet.

Hope this helps somewhat with the 'wrapping'. Shout if something is unclear, :)

-J

The spacetime curvature center discussed here is of the local variety
I thought it must be.
What we do know is that when we observe the effects of gravity, we also observe the slowing down of clocks
I can visualize planets orbiting around the sun as moving in curved space. I can't visualize slowed-down time as curved time.
Instead of saying "time is passing", maybe it's better to say "we are passing through time" (i.e. as traveling in space). This viewpoint suggests that possibly not all parts of the universe are "traveling" at the same rate, just as some galaxies are moving apart, and some are moving together. One 'bubble' may be attracted to another 'bubble'. Since the universe is far bigger than we can see, we may never know.

For those of us still struggling with accepting the Standard Model, there is yet hope. Seems there are some serious issues with how the Standard Model predicts the growth and distribution of galaxies in the Universe:

http://www.newscientist.com/article/mg21028161.300-slim-and-beautiful-galaxies-too-good-to-be-true.html?DCMP=NLC-nletter&nsref=mg21028161.300

Note that the link to New Scientist may require a log-in.

Hi Jorrie,

In your note #3 you provide dtau2=(1-rs/r)dt2 - dr2/(1-rs/r) as the schwarzschild metric.

I was reading the following link:

http://www.rafimoor.com/english/GRE3.htm

And was interested in this part:

The Interior Solution

A second solution given by Schwarzschild was an approximation to the metric inside a star. Schwarzschild assumed a static spherical body made of incompressible perfect fluid. This case is also spherically symmetric and static but its stress-energy tensor is nonzero. It has nonzero pressure and density at every point inside the star. The density is constant at all the space inside the star (this is what incompressible means) and the pressure is only a function of the radial coordinate r. The line element equation for this metric is:

(3)

for r ≤ R , where R is the radius of the star.

We can see that if we substitute r = R in (2) and in (3) we get the same result, so that there is continuity between the interior and the exterior solutions.

My Question

Making the same assumptions as they did in the link, but for the 2 dimensional spacetime, what would be the interior solution? Just the c²dt² and dr² terms unchanged?

Hi Roger,

I've been a 'tourist' last week, with only sporadic internet; hence the silence. :)

"Making the same assumptions as they did in the link, but for the 2 dimensional spacetime, what would be the interior solution? Just the c²dt² and dr² terms unchanged?"

Yes, I think so. For 2-D spacetime, there cannot be any dθ or dφ in either interior or exterior solution.

-J

Hi Roger.

That old paper is really cool! I haven't noticed it before, probably because my interest lies more in the exterior and the cosmological solutions. However, this one:

seems much more general than the incompressible solution. It may have some bearing on my quest to understand the 'bulging effect' of fast spinning planets with non-linear density profile. I will check what it looks like in the weak field approximation and how I can possibly utilize it...

Thanks!

-J

Jorrie, if you find the time, perhaps you could take a peek at the question I posted yesterday regarding the Cartwheel Space Station. Your opinion would be highly regarded and appreciated.

I was thinking about a practical measurable application of the schwartzschild interior solution and I thought of the fluid dynamics of the cores of stars and planets. It would be interesting to see the effects that GR time dilation would have on convection currents within a star (My first and probably wrong guess is the currents would be more elliptical). When I have more time I'll search for a paper on the subject (I'm sure there are thousands on this).

Hi Roger,

Interesting point, but my knowledge on astro-physics is extremely limited.

From a 'relativist' point of view, the effect is probably "weak-field" in ordinary stars and not very important, but I'm not sure...

-J

I don't know enough to know whether you're right or not about the weak-field approximation (I'm sure you probably are). But I did find a super cool presentation on General Relativistic Hydrodynamics that almost made my head explode trying to understand it. I will have to reread it many times to understand 10% of it, you'll undoubtedly have better luck.

I had to pass it along, if only because of how awesome it is.

http://www.uv.es/jofontro/TALKS/Font-Playacar.pdf

Hi Roger.

I agree - very impressive document. But, I'm afraid the math is a tad too impressive for me.

I see that they calculate all sort of stellar sources of gravitational waves, which may perhaps (IMO) be observable by a spacecraft in close solar orbit, or maybe even by the future LISA space detector...

-J

I feel bad that there was only one among us that could have an intelligent conversation with Jorrie about that.

I feel mentally retarded at this point.

Question: If mass bends space-time and the space-time is bent by the Sun and is static, what does the Earth do to the space-time bent by the Sun as it orbits the Sun and the Moon do the space-time bent by the Sun and Earth as it orbits the Earth? The static rubber sheet analogy only works because the only mass shown bending space-time is the central mass. Orbit another mass around this which MUST in turn bend the space-time already bent around it and it becomes untenable that it can be static. Since according to SR it is bent space-time and not gravity which causes orbits, then the earth must indeed be bending space-time around it for the moon to follow, bending space-time already bent by the Sun. As the earth orbits the Sun space-time must bend and unbend as it passes, there can be no static space-time. Likewise the Sun must bend space as it orbits the galaxy which bends the space around it. Static in such a situation has no relevance.

That's another problem I have, gravitational waves. SR is bending of space-time, no gravity at all, merely bodies in free fall following a curvature of space-time. So is it bent space-time that controls planetary orbits or gravity? The two are mutually exclusive. To date, LIGO has found no gravitational waves from any source where they were expected to be found.

In a one article They laud success from complete failure:

[quote]"LIGO's contribution to the study of GRB070201 marks a milestone for the project, says Caltech's Jay Marx, LIGO's executive director: "Having achieved its design goals two years ago, LIGO is now producing significant scientific results. The nondetection of a signal from GRB070201 is an important step toward a very productive synergy between gravitational-wave and other astronomical communities that will contribute to our understanding of the most energetic events in the cosmos.""[/quote]

By detecting nothing where they were sure they existed they have made significant scientific results.

Another story (http://news.ufl.edu/2009/08/19/gravitational-waves/ - sorry, link no longer available) again lauds success from failure:

[quote]"Gravitational waves are the only way to directly probe the universe at the moment of its birth; they're absolutely unique in that regard," said David Reitze, a UF professor of physics and the spokesperson for the LIGO Scientific Collaboration. "We simply can't get this information from any other type of astronomy. This is what makes this result in particular, and gravitational-wave astronomy in general, so exciting."[/quote]

[quote]"Since we have not observed the stochastic background, some of these early-universe models that predict a relatively large stochastic background have been ruled out," said Vuk Mandic, assistant professor at the University of Minnesota and the head of the group that performed the analysis. "We now know a bit more about parameters that describe the evolution of the universe when it was less than one minute old."[/quote]

It has indeed ruled out models relying on gravitational waves, perhaps even the very theory that predicts them? The question is how many more billions of your tax dollars are you willing to spend on a project that has detected nothing from its inception and then attempts to turn that detection of nothing into success in order to obtain more funding?

Is it lack of sensitivity? According to the LIGO fact page:

[quote]"Gravitational waves are ripples in the fabric of space-time. When they pass through LIGO's L-shaped detector they will decrease the distance between the test masses in one arm of the L, while increasing it in the other. These changes are minute: just 10^-16 centimeters, or one-hundred-millionth the diameter of a hydrogen atom over the 4 kilometer length of the arm. ... "

[bold emphasis mine SJW][/quote]

Ok, take the diameter of one hydrogen atom, (already a really really small thing) divide it into one hundred million really really little bits, and one of these bits is the change in length over a 4000 m distance that a gravity wave will cause... Yet nothing.

Hi SJW, I've just completed my GPS Blog article and do not have the time now to reply to the gravitational wave issue.

Just one thing in your opening statement above: SR does not bend spacetime; that's done by GR, when there is mass present. SR can deal with acceleration, but not with gravity.

Yes, there are special cases where SR is a good enough approximation of GR, even in the presence of massive bodies, but this is not valid in general.

-J

Yah, I tend to get those mixed up being both are in error anyways, but since one is just a generalization of the other, what should it matter? One bends a space-time that is not an aether and is therefore composed of nothing and this nothing then tells mass how to move, sounds logical to me. Wait, what? Of course we tend to gloss over the fact that something first affects nothing and then nothing somehow affects something, but don't let it bother the layman too much. It's simply beyond his mathematical understanding. In other words don't worry about it, it's true and that's all you need to know. good snake oil propaganda they have. So either space-time is not an aether and so is composed of nothing, or it is something in which case it is an aether. A nothing affected by somethings and then in turn affecting other somethings may not bother you, but last I checked nothing still equaled zero.

I will be interested in hearing your reply to gravitational waves when you find the time.

"So is it bent space-time that controls planetary orbits or gravity? The two are mutually exclusive."

Nope. I can add another one: 'compressed spacetime'. All three are valid views - take your pick of the most convenient one for the job at hand.

"Is it lack of sensitivity?"

For gravitation waves, sensitivity and bandwidth are issues. LIGO has achieved about as much sensitivity as is feasible on the ground, in the limited frequency range 2 to 1000 Hz. So far nothing within that range. Maybe never, until LISA, the space based g-wave antenna is eventually in place.

The process of science is to formulate one or many theories that explains all or most of what has already been observed and quantified. Then use those theories to predict what has not been observed (yet). As soon as any theory looks credible, explains all past observations (within the error bands of results), and can make some new predictions, then push technology and engineering to enable a future measurement. Any good measurement, even a null result, is good for science, because it restricts the theories that are feasible.

GR is just one such a theory. G-waves is a definite prediction of GR. Its effects have already been observed indirectly in binary pulsars, but the amplitude and frequency are a little uncertain for those conditions. Events that create the predicted g-waves of sufficient strength are (fortunately for our health) quite rare and hence sparse. The more sensitive and the wider the bandwidth of detectors, the larger the volume of space it can cover. So, while using the present generation of detectors are looking out, new ones are waiting to be built.

If I were a betting man, I would not have risked my money against future direct detection of g-waves on or around earth...

-J

PS. You can read more details of what I know on the subject on my website, linked to in the Blog header above. Check for the pdf under General relativity, Gravitational Waves.

I will take that bet and predict that LISA detects nothing any more than LIGO did. The reason you need gravitational waves is you refuse to take the E/M force into account. A child with a magnet can pick up a steel ball against the entire gravitational pull of the earth, yet you give no account to the E/M force acting in space. Even in a plasma where only 1 in every 10,000 particles is charged the E/M force is 10 to the 7th power stronger than the gravitational force, yet you discount it in every explanation.

So tell me, what is space-time composed of that allows it to be bent or compressed?

"So tell me, what is space-time composed of that allows it to be bent or compressed?"

"or expand", you should have added.

Easy, you just mix space and time, let it stand for ≈ 10^-43 seconds and "bang", you have spacetime.

Frankly, what each consists of, nobody knows. Do you know what time consists of?

Further, do I understand correctly that you say it is the electromagnetic force that keeps earth in orbit around the sun and also the force preventing us from all drifting off into space?

-J

Yet ask any relativist what space is and it will certainly not be described as composed of anything. What is space-time expanding "into" if not well, space? Yes, time is composed of a cesium atom and it's movement from one place to another as it vibrates back and forth, or the distance the earth revolves around the Sun to reach the same point, or the distance the earth revolves around it's axis to reach the same point, or so many swings of a pendulum. It is man's attempt to quantify movement. There is no past or future in the universe, only the here and now. Man has a basic need to try to explain everything around him, time cannot be explained because it does not exist except in the mind of mankind. Time is no more a separate dimension than distance is, it is just another form of measurement to measure distance. it is not time that changes, but the device itself that measures it when influenced by gravity or acceleration, it is therefore not a property of space, but a property of the cesium atom you use to measure it. The entire universe is E/M in nature, one would think the photon might give you some idea that this is the basic structure of everything. It is not coincidental that photons are E/M in nature, or that E/M radiation is emitted by everything that exists.

No it is the electromagnetic force acting in concert with gravity that keeps the earth in orbit around the Sun and also prevents us all from drifting off into space. It is neither one nor the other but a combination of both working in harmony. The magnetic force provides the spin and tangential motion that gravity cannot provide, a simple fact that magnetism causes things to circle perpendicular to the electrical force, hence all spin and orbits follow the right hand rule. I do not claim that gravity is not involved, nor do I ignore one of the strongest forces that exist, a force emitted by every single particle that we know of. The E/M force cannot explain planetary or galactic orbits without gravity anymore than gravity can do so without the E/M force, unless of course you invoke mysterious undetectable substances to do the work of the E/M force and apply the name gravity to it.

"Yes, time is composed of a cesium atom and it's movement from one place to another as it vibrates back and forth, ..."

Take note that this is not how cesium clocks determine time...

Anyway, you have not said anything on what time actually is, just how we defined it for measurement purposes. That's OK with me, because nobody can do any better.

"... it is the electromagnetic force acting in concert with gravity that keeps the earth in orbit around the Sun and also prevents us all from drifting off into space."

I then presume you can quantify the contributions of the e.m. and gravity components of the force for us? If possible, do so for both 'us' being kept on the ground and for earth being kept around the sun. If you can do that, it will be a huge contribution to this Blog.

-J

Being no expert I can only give you a layman's perspective. Charge differentials. Just as an electron is held in orbit by the E/M force due to charge differentials, it neither approaches too near the nucleus nor does it drift off into space, even with a velocity close to c. (Another problem with infinite energy required to reach c, but well get to that later.)

According to gravitational theory gravity is almost non-existent at the atomic level, yet you would have it by itself hold stars, light-years apart from drifting off into space as a galaxy rotates. It is a simple fact that electrical forces both attract and repel, this is experimentally proven and I doubt anyone will disagree and that the magnetic force causes things to circle perpendicular to the electric force. Most will claim space is electrically neutral, but this is not true. Why does a spacecraft when launching into orbit build an electrical charge on its surface, if space is electrically neutral? It builds a positive charge on the sunward side and negative charge on the shadowed side. Clearly Earth contains a charge difference different than both the space around it and the Sun.

As a planet is pulled towards the Sun by charge difference, remember, two unlike charges attract, it begins to accumulate the same charge as the Sun and remember, two like charges repel. This is why planets neither spiral into the Sun, nor drift off into space, but most indeed travel in an elliptical orbit. Gravity of course only attracts, or is it simply a misunderstood component of the attraction of the E/M force? That has yet to be discovered, since the source and transmission of gravity is unknown.

Yet clearly a child with a magnet can overcome the entire gravitational pull of the earth, so attributing planetary orbits only to such a weak force is illogical. Add the attractive and repulsive aspects of the electromagnetic force and the problems with a gravity only model disappear.

Yes, we are held on earth mainly by the gravitational affect, but then being basically part of the earth the charge differences between us and the earth are not very great. Composition of planets play a large role as well in their ability to hold charge, therefore determining their orbital distance from the Sun, just as electrons with higher charge change orbits from lower ones. Gravitationally larger planets should orbit closer, given the larger attraction between the two. Yet we find the larger planets orbiting at greater distances, as they can hold greater charge having more mass and likely composed of material able to maintain a greater negative charge.

We can extend that to comets and the data from Deep Impact if you want to discuss formation theory as well.

Hi SJW, perhaps over-speculative, but not a bad effort.

I will defer most of it to readers who know much more about sub-atomic physics than me. I will however comment on gravity statements that I consider suspect.

"Gravitationally larger planets should orbit closer, given the larger attraction between the two."

No, just faster or farther, not closer. Standard orbital dynamics.

Actually, the electrical charge of a planet does add a little to its rest mass and hence to its gravity, but the effect is minuscule.

You realize that your theory flies into the face of Newton's theories as well as Einstein's? Such exceptional claims require exceptional evidence.

-J

We have already discovered the currents connecting Sun and earth which effect is the aurora, began mapping them with THEMIS. Jupiter and Saturn display these same effects between planet and moons, so it would be illogical to assume the cause is completely different.

Yet Flat galactic rotations can not be explained by the same standard orbital dynamics, so instead of examining the underlying theory to see what is wrong or what is missing, they make up an undetectable particle, name it Dark Matter, and place it precisely where they need it. Supposedly it only interacts with mass when they need it to, and ignores it in all other circumstances. Problem solved, but of course since they had to add a mysterious ad-hoc force they now create another problem so we have to invent Dark Energy to counteract the Dark Matter so the universe can still expand. The E/M force both attracts and repels, DM and DE all in one thing, coincidence? It is also testable. Then when experiments showed Cold Dark Matter was no longer viable instead of once again examining underlying theory they simply change its temperature to Warm Dark Matter. Why such an aversion to the E/M force, when the entire theory of Einstein relies on the fact that a photon is the E/M force? Without the E/M force his theory would have no basis to stand on, would in fact be no theory at all. One would think you would embrace it, not make up undetectable entities to try to explain it away. He was way ahead of his time, it's only too bad he died before he could revise his theories with modern day data.

If our universe (is uniform, and) has a "closed" global curvature, then (constant-time) space "slices" of our space-time fabric would be hyper-spheres. Omitting one spatial dimension; and, the time dimension; then such a hyper-spherically curved-and-closed space-time fabric could be (approximately) visualized, via "(2+0)D in 3D", as a "sphere-land" (cp. "flatland").

^{(source: world mysteries)}

The above visualization is valid, strictly speaking, only for an idealized, i.e. uniform, universe, where matter is "smeared evenly throughout space", like a "creamy peanut-butter". But, our actual universe is not homogeneous. Instead, our actual universe is "clumpy", like a "chunky peanut butter". What is the effect, of these anisotropies, in the distribution of matter, in our universe ?

In his book Journey into Gravity & Spacetime, physicist John Archibald Wheeler emphasized, that General Relativity, is an entirely "local" theory :

• matter "here" tells space-time "here" how to curve
• curved space-time "here" tells matter "here" how to move

Therefore, if you could cause matter, in an initially-uniform universe, to "clump"; then, the initially-smooth space-time fabric would warp, to accommodate the now-non-uniform matter distribution, i.e. the "new instructions", input into the space-time fabric, from the newly re-distributed matter, inside that space-time fabric. Therefrom, "sphere-land" would become "poly-hedron-land" -- i.e. approximately spherical, globally (on large scales); but, flat, locally (on small scales), in the empty "voids" between the "clumps" of matter (cp. circles can be approximated with polygons):

^{(source: John Zerning)}

Indeed, our cosmos "clumps" into a "Cosmic Web", which Large-Scale Structure resembles the frame-work, or mesh, of a (hyper-)poly-hedron:

• Clusters _<---> vertices
• Filaments _<---> edges
• Voids _<---> faces
  

^{(source: Berkeley)}

If so, then space-time is "robust", i.e. the global solutions, generated by General Relativity theory, on large scales, are not radically altered, by local non-uniform (non-radical, cp. NS & BH) re-distributions of matter-and-energy. Qualitatively, GR is an entirely "local" theory, according to which matter "here" can only (directly) affect space-time "here" (although warpings "here" can be communicated, "through the gossip grape-vine", i.e. via gravity-waves, to "there"). Thus, the local "stage" of space-time is "set", by its surrounding space-time environment, i.e. global "big picture" on larger size scales.

In the remote universe, at a red-shift of z~1000, the CMB is structured, with a maximum amplitude (~10^-5), at angular-size scales of ~0.7 degrees (UCLA), corresponding to a current co-moving spatial-size scale of ~100 Mpc (Caltech). Meanwhile, in the local universe, at red-shifts z < 0.2, luminous matter is structured, with a maximum amplitude, at spatial-size scales, of ~100 Mpc (Landy 1995, Scientific American).

Thus:

• CMB structure "there & then" correlates to Large-Scale Structure "here & now"
• CMB structures, which were "there & then" ~100 Mpc / (1+z) in spatial-size, have evolved into LSS structures "here & now" ~100 Mpc in spatial-size, i.e. CMB structures have stretched ~1000x with expanding space-time
• "weak" CMB structures, which were "there & then" ~10^-5, are predicted (by linear perturbation theory) to have "strengthened" by a factor of (1+z) ~ 1000, to be ~10^-2 "here & now"

So, "weak" CMB structures are predicted to have passively stretched with expanding space-time; and, to have evolved into LSS, i.e. 'Cosmic Web', representing merely "minor ripples" ~1% on an otherwise nearly homogenous ambient matter distribution, at present epoch.

These predictions are completely consistent, with known mass & volume distributions, of 'voids' vs. 'structure' (i.e. 'filaments' & 'clusters'):

• 'voids' = 76% volume, 74% matter (i.e. 'Dark Energy')
• 'structure' = 24% volume, 26% matter (i.e. 'baryons' & 'Dark Matter')

Therefore, despite the "optical prominence" of the CW, such LSS represents an as-yet-weak perturbation ~1%, in the nearly homogeneous matter distribution, in our universe. Compared to current cosmological computer simulations, which seek to "pigeon hole" all matter, into shapes similar to the "optically prominent" CW, actual LSS represents an as-yet-weak "clumping", of the ambient Inter-Galactic Medium. That "clumping", resembling current cosmological computer simulation initial conditions (see figure below), actually represents merely a "weak" over-density ~1%. However, actual LSS does represent a "dramatic" re-distribution of matter, within the regions of space occupied by the CW, i.e. the "compactification" of warm diffuse IGM (i.e. primordial HII) into galactic structures (i.e. galaxies & stars), i.e. "phase change at (nearly) constant density". Dynamically-young, primordial IGM has been observed (Discovery).

^{(source: Carnegie Mellon)}

You are using a lot of acronyms here; few outside the field will be familiar with them, so please define. Your post is quite hard to follow; please try and be more descriptive and perhaps put less info into one post.

I may help if you can give us a summary of what you are trying to convey here and how it connects to spacetime curvature.

Beware "by analogy" ?

The actual curvature caused, by a mass M, into the fabric of space-time, scales with that mass' gravitational radius R_S. For our earth, R_S ~ 1 cm.

Now, our earth is ~10,000 km across. Yet, in most "rubber sheet" visualizations, our earth is represented by a bowling ball, ~10 cm across -- i.e. a reduction in scale by eight orders-of-magnitude. Meanwhile, the "sag" of the space-time fabric, is actually exaggerated, visualized as ~10 cm or more. Thus, thereby, our space-time fabric is perceived as "soft & squishy". Yet, on the same reduced scale, as the "bowling ball earth", the "rubber sheet" should "sag" only 1 nm, i.e. little more than one atomic width.

So, our space-time fabric, far from being "soft & squishy", more closely resembles an ultra-strong armor plating -- which, when "loaded" with a world's worth of mass, "sags" a single centimeter. Qualitatively, then, detecting gravity-waves would be more akin, to hearing the "twang" of steel drums (or, Peter Frampton's "talking guitar" in Do You Feel Like We Do ?); than to a "thrumming" of un-stretched guitar strings.

In particular, then, according to GR, "gravity is not weak"; instead, "space-time is strong". Thus, the observed effects of gravity are weak, for the same reason, that ping-pong-balls rolling across the armored deck of a battle-ship, would produce imperceptible impacts upon that metal.

Could you generate a region of energetically "over-loaded super-critical space-time", which was full of "latent energy", in the form of photons, which destructively interfered, e.g. two high-intensity but oppositely-directed laser beams; and, then, rapidly re-adjust the photon field, until all the photons were suddenly "in phase", and constructively interfered? Such a rapid "spike" in the local energy density could, conceivably, "pop" space-time, like a hammer tapping on a steel-drum.

The problem with space-time curvature is it doesn't work without gravity, plain and simple. Nor does it work unless space-time is made up of an aether. You can not curve nothing with mass and have that nothing affect mass.

If gravity is now defined by curvature rather than by a centripetal force, what impels an object placed at rest in a field to begin moving? General Relativity supplies us with field differentials, which can explain why an object already moving in the field will move as it does. But field differentials, being math, cannot create a force. The math of GR represents motions, it cannot cause them. GR is also not a field of potentials, since it requires a field of forces to create potentials. GR is not a field of forces, so the differentials cannot be interpreted as potentials. Einstein admitted that GR was the bypassing of Newton's inertial field. How can an object that is "feeling no forces" begin moving in such a field? In other words, Einstein inherited and extended the field of Newton, but he did not overwrite Newton's first law. If he had, we would not still be taught it in high school. Newton's first law is that an object at rest will remain at rest unless a force acts upon it. What force acts upon an object placed in Einstein's curved field? How does the object know that the field differential just below it is any different than the field differential it inhabits? It can't know, and therefore GR fails to explain motion from rest in a field.

An isolated object in free space can not be said to be at rest or in motion. One needs at least two objects to define motion. I also believe that applies to gravity as well. An isolated object can not be determined to have mass. You at least need a photon in the vicinity to determine if the space is actually curved. If your object and the observer are sufficiently separated such that any gravitational effects between the two cannot be detected, how do you propose the observer determine the mass?

If we assign an arbitrary mass to our isolated object, we still can not determine if space time curves about the object unless we have the minimum of a photon passing near by. Essentially, an object can not be defined or characterized except "relative" to some other object...

Stick two objects in close proximity in Einstein's curved space and there is still no reason for either to move. Remember in Einstein's curved space there are no external forces acting, no reason for one object to begin rolling down the slope caused by another. There is no down or up, nor any reason for the two objects to approach. His equations are nothing more than equations that describe how bodies move in a gravitational field, but without the force of pre-existing gravity there is no force to start the movement the equations describe. In reality there is no curved space. His equations may adequately describe motions in Newton's gravitational field, but it in no way as Eienstein said the bypassing of Newton's inertial field.

"His equations are nothing more than equations that describe how bodies move in a gravitational field, but without the force of pre-existing gravity there is no force to start the movement the equations describe."

Not quite. Have you studied Einstein's field equations?

Note the words: "influences ... movement ... through spacetime." All things move through spacetime all the time.

I understand, that General Relativity imputes an "ontological existence", to the space-time fabric, i.e. "the stage" of space-time is as real, as "the actors" -- i.e. massive objects, whose masses cause "the stage" to "sag", affecting every "actor's" movements, as they all "dance around on stage".

Simply qualitatively, to "order-of-magnitude", eq. 1.9 implies that:

R [length^-2] ~ T [energy length^-3]

Now, energy ~ mass ~ length, i.e. "Schwarzschild radius". So, the qualitative magnitude, of the curvature [length^-2], imputed into the space-time fabric, by a distribution of matter-and-energy, scales as the gravitational size per volume [length length^-3]. Or, inversely, the radius-of-curvature [length], of the warped space-time fabric, scales as the (square-root of the) volume-per-gravitational-size [(length³ length^-1)^1/2].

Note, that the "smearing" of particles, at Quantum Mechanical size scales, "saves" the space-time fabric, from the "singularities", of mathematically ideal point-particles. Brashly extrapolating, perhaps some qualitatively similar effects "smear" out the "singularities", in black holes ?

the Schwarzschild solution sets the Ricci Tensor to 0 which means there is no mass in the universe. It was later corrupted by David Hilbert who added mass ad-hoc without first describing it by an energy momentum tensor. I would be willing to bet the solution you ascribe to Schwarzschild is not even his formula, but the corrupted version by Hilbert.

In mathematical terms, those solutions obtained by Schwarzschild, Droste and Weyl, and Brillouin, are mutually consistent, in that they can be obtained from one another by an admissible transformation of coordinates. However, Hilbert's solution is inconsistent with their solutions because it cannot be obtained from them or be converted to one of them by an admissible transformation of coordinates.

Brillouin demonstrated rigorously that the procedure of Hilberts solution leads to a non-static solution to a static problem, just as Droste had pointed out in 1916, contradicting the very statement of the initial problem to be solved - what is the gravitational field associated with a spherically symmetric gravitating body, where the field is unchanging in time (static) and the spacetime outside the body is free of matter (i. e. vacuum), other than for the presence of a test particle of negligible mass? So Schwarzschild's solution is of a single gravitating body in a universe devoid of all other mass.

To claim Einstein's field equations refer to two or more bodies can be done in only two ways:

1. Derivation of an exact solution to Einstein's field equations for the two-body configuration of matter; or

2. Proof of an existence theorem.

To date none exist.

"the Schwarzschild solution sets the Ricci Tensor to 0 which means there is no mass in the universe."

Nonsense. The Schwarzschild solution:

You notice that "mass M, at rest at the origin of the coordinate system"?

That is not Schwarzschild's solution, you are indeed using Hilbert's corrupted version just as I figured you would be.

Here is Schwarzschild's paper: http://www.sjcrothers.plasmaresources.com/schwarzschild.pdf

His equation is (14)

Here is Hilbert's equation: http://www.sjcrothers.plasmaresources.com/hilbert.pdf

Notice how he adds mass ad-hoc into the Schwarzschild solution and calls it Schwarzschild's solution.

Here is Brillouin's paper which agrees with Schwarzschild: http://www.sjcrothers.plasmaresources.com/brillouin.pdf

Another paper by Schwarzschild: http://www.sjcrothers.plasmaresources.com/Schwarzschild2.pdf

Only Hilbert's solution is corrupted by adding mass ad-hoc and then called Schwarzschild's solution. Just as I figured you had no idea the solution you attribute to Schwarzschild is not even his solution.

Dear, Oh Dear.

You ever noticed his definition of constant alpha?

I rest my case.

And tell me, what does the Ricci Tensor say that mass is? 0.

Dear, oh Dear.

"And tell me, what does the Ricci Tensor say that mass is? 0."

Of course not!

The Ricci tensor R_μν is not telling us about mass, but about curvature; mass-energy distribution is represented by the energy-momentum tensor T_μν. In the case of Schwarzschild spacetime⁽¹⁾, the sum of the components of T_μν=0, but all 16 components are not zero. They simply cancel out.

If you do not understand that, please get a good book on the Schwarzschild solution.

-J

(1) Schwarzschild spacetime exists around a gravitationally uniform, symmetrical, isolated, non-charged and non-rotating mass. It is the simplest special case and does not describe the real cosmos. Einstein's field equations represent any distribution of masses, moving in arbitrary fashion relative to each other.

Read through the first link... specifically what you pointed to... right under eq (14)

The latter contains only the constant alpha that depends on the value of the mass at the origin.

W: "Brashly extrapolating, perhaps some qualitatively similar effects "smear" out the "singularities", in black holes ?"

Quite possibly something like that. The quantum gravity guys are trying hard to put it on a consistent footing for us.