Upgrading Newton's Universal Gravity Law
By Doug Sweetser | December 20th 2013 08:11 AM | 34 comments | Print | E-mail | Track Comments

Trying to be a semi-pro amateur physicist (yes I accept special relativity is right!). I _had_ my own effort to unify gravity with other forces in...

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[note: this blog, as written is insufficient. I will need to provide an action to clarify the nature of the proposal.]

The Spoon and the Moon

Give a baby a spoon, and the little one is likely to toss it to the floor, greatly amused.  Return the spoon and the baby will gleefully demonstrate that the experiment can be repeated.

The Moon falls around the Earth.  The Moon has not stopped falling toward the Earth for four billion years.  It may be the greatest accomplishment of abstract scientific thought that the same law that governs the spoon falling also governs the motion of the Moon.  Newton's Universal Law of Gravitation was the first expression which established that the same law works in the heavens as it does on Earth.  Here is that law:

$m \frac{d^2 R}{d t^2} = - G \frac{m M}{R^2} = - m \nabla \Phi$

The small m could be the mass of the spoon or the Moon.  Notice it appears on both sides.  Written on the left, the mass times acceleration is an inertial mass, a resistance to change.  The spoon is easier to move around than the Moon.  On the right, there is a minus sign which indicates the force is attractive.  The gravitational attraction of the small m to the big M (Earth) must be divided by the distance squared.  The gravitational constant G is there so that the expression has units of force.

The inertial mass m happens to be the same as the gravitational mass m.  In perhaps the second greatest accomplishment of abstract scientific thought, Einstein used that equivalence to construct his theory of gravity.  I will not try to explain that idea in any detail other than to say it is about dynamic space-time geometry, where something is in a gravity field changes measurements of both time and space.  General relativity is like a sublime wine, the more opportunities one gets to study the subject in detail, the more giddy one gets with its elegance.

The Problem with Light

The effect of gravity is universal.  Gravity even tugs light.  Watching the position of the stars during an eclipse gave the first data that the path light of the heavens is altered by the mass of the Sun.  Using radio astronomy, it was possible to study light bending at any time of the year.  All the data that poured in was consistent with the predictions of Einstein's approach to gravity, general relativity.

There is a problem with Newton's Universal Law of Gravity written down by Sir Isaac himself.  The mass of a photon is zero.  The law applied to photons says zero equals zero.  While true, it is no use for doing a calculation.  Physicist know to use general relativity.  Yet I think one should give the father of physics more respect.  Is there an upgrade one could apply to this grand old law?

Why not replace the mass of the spoon or the Moon on either side with universal constants? There is a combination of such constants known as the Planck mass (the weight of a flea egg according to wikipedia).  Let that be a stand in:

$\sqrt{\frac{c \hbar}{G}} ~\frac{d^2 R}{d t^2} = - \sqrt{G c \hbar} ~\frac{M}{R^2}$

A photon with a mass of zero will not be a problem for this law.  The upgrade make's Newton's law more universal because it can also be used to describe the motion of light.

Physicists have already calculated how much Newton's theory predicts light bends in the limit of the mass approaching zero.  The theory gets half the amount seen.  This upgraded gravity law remains inconsistent with experimental data and so remains wrong.

The upgrade is interesting for a number of reasons.  The law now looks like it is only about geometry [note to the technical reader: out of habit, I wrote "space-time geometry", but that is not right since this upgraded law remains fixed to Galilean geometry].  On the left are a few constants and an acceleration, the change in the change of space per unit time.  Upgrading Newton's theory shows respect for the accomplishments of general relativity: gravity is only about geometry (albeit the wrong geometry).

General relativity was born from a careful analysis of the equivalence of inertial and gravitational mass.  Now neither is in the equation.  Interesting.

Newton's original equation had one constant, G, whose value was not known to Newton.  The presence of the Newton's gravitational constant indicates the equation is related to gravity.  The field equations that make up general relativity have the constants G and c, the speed of light.  This indicates the equations touch on gravity and special relativity.  The upgraded gravity law brings in a third constant, Planck's constant, hbar.  That is the calling card of quantum mechanics.  On dimensional grounds, the upgraded law looks like its domain of application should be relativistic quantum gravity, the driver of so much research today.  I am NOT claiming to have a full blown quantum gravity model.   I don't even have a toy model.  I think this is worthy of more investigation, and that's about it.

The solution to the field equations

Field equations go hand in glove with the equations of motion.  The classical field equations gravity look like so:

$\rho = \nabla^2 \Phi$

The solution for a point charge is:

$\Phi = k ~\frac{M}{R} + C$

There are two integration constants because this is a second order differential equation.  Take two spatial derivatives of M/R and the units are mass per volume, so k is dimensionless.  Is k positive or negative?  The force equation must be attractive:

$F = - \sqrt{\frac{c \hbar}{G}} ~\nabla \Phi$

[note: out of habit, I started to write "like charges attract", but realized with the upgrade to Newton's law proposed here, there was only the gravitational mass.  Getting rid of a cliché phrase, how fun is that?]

A spatial derivative of M/R will bring in one minus sign, so k must be negative so that the three negative signs end up staying negative, and the force is attractive.

The negative k creates a problem: the field has a negative energy.  The problem can be avoided if the additive constant C is HUGE.  Making certain the energy of the gravitational field was positive definite was a problem that bothered James Clark Maxwell (see an excellent page devoted to the subject written by Kevin Brown).  Maxwell was unable to justify creating such a huge positive value.  In general relativity, the metric term g00 is one minus a super-small M/R term due to the constants G/c2, a small number divided by the square of a large number.  Use the inverse of those same constants to make the energy in the field positive definite but the gradient negative:

$\Phi = \frac{c^2}{G} \left(1 - \frac{G M}{c^2 R} \right)$

It is quite common for people to ignore the additive constant.  Yet it does solve a problem the perplexed Maxwell, so is worthy of noting.

Limits to the Upgrade

Newton's universal law of gravitation is now more universal because it applies equally well to spoons, moons, and photons.  Just because it applies to photons does not mean it gets the answer right: it is off by a factor of two.  On a deeper level, the upgrade does not make Newton's gravitational law about space-time geometry.  Space remains separate from time via Galilean transformations. The speed of light is still infinite according to the field equations.  Newton's gravity law only changes time, not space, and again not consistent with data.  Wrong, wrong, wrong.

The upgrade is interesting to me.  I've been saying that like charges attract since I first learned of gravity.  Now I cannot say that.  Gravity is not about charges in upgraded Newtonian theory which is a good thing.

I can no longer write the Universal Law of Gravity as Newton did at the dawn of the modern scientific process without seeing the photon zero equals zero blunder.  Wrong is wrong.  Unless I am wrong about the upgrade to the oldest gravity law on the books.

Let me restate what you just did, to help you understand some problems here:

You took as a starting point

From this point of view, it looks like the 'm' is spurious and can be ignored unless m=0 (which it is then unclear what happens).  So you decided to get rid of any 'm' by replacing it with a universal constant (you chose the planck mass, but you could have chosen 1 kg, or anything you felt like).  Now there is no m, so saying m=0 means nothing at all to the equations.

So what is wrong here?  Or alternatively, why is there an m at all in the first place if it has no effect?

*Spoiler below*

If mass didn't affect anything, then we couldn't even measure it.  So what went wrong is that you were neglecting where those equations came from.  We have Newtonian mechanics, and we have a force law for gravity.  Together they give that equation.  So while you meant to just tweak gravity, you actually tossed out all of Newtonian mechanics.  You changed:

to just

So now all of Newtonian mechanics (except when only gravity is involved) will no longer work.

Also, despite tossing the mass of one object, you still leave in the mass of the other object in the gravity force equation.  How does the universe decide which masses to ignore in this idea?  In your case, you chose to get rid of all inertial masses, and keep all gravitational masses.  So you are treating the two very differently, while inertial mass just becomes a universal constant the gravitational mass still shows up unblemished in the gravitational potential equation.

So you are extremely violating the equivalence principle now.  Kind of ironic since it sounds like that was an initial motivating factor for you.

Some other things:
out of habit, I wrote "space-time geometry", but that is not right since this upgraded law remains fixed to Galilean geometry
Because the equation of motion for a free massive particle in Newtonian gravity is independent of the mass, it's like the particles all feel the same curvature.  Indeed this can be expressed instead as curved space-time.  While it is not the usual Lorentzian space-time, I see no problem calling it a space-time.  It is just a Galilean space-time. (I have seen people use that phrase.  Often when discussing Newton-Cartan theory.)

The speed of light is still infinite according to the field equations.
I know what you are trying to say, but just to be clear: The Maxwell field equations still give a speed of light as finite.  There is no speed limit in Newtonian mechanics (and the gravitational field in Newtonian mechanics does propagate changes instantaneously), and this is like the "c" parameter in Lorentz symmetry going to infinity to get Galilean symmetry.  But the "c" parameter which was sent off to infinity there is no longer connected with the speed of light.

Is there any experiment in classical mechanics that allows someone to tell the difference between the inertial masses of the spoon, the moon, and a photon?  It may well be that every such experiment allows one to tell the difference.  I can toss a spoon against a wall.  I cannot change the motion of the Moon by pushing against it since it has far too large an inertial mass.

This blog was not about inertial masses in classical mechanics.  This blog was exclusively about the fundamental force of gravity.  With a fundamental force, no one does a damn thing, and the fundamental forces do their own thing.  That is the subject at hand.  I am guilty of dealing with gravity only.

Go to the Tower of Pisa with two masses, one three kilograms, the other five kilograms.  They were carefully designed to have the same amount of drag.  What experiment using only gravity will allow someone to tell the difference between the two?  According to general relativity, the spacetime curvature is due to the stress-energy tensor of everything, including these two masses.  They will go into free-fall motion following geodesics.  An experimenter will not be able to say - based only on this experiment - which mass is three kilogram or five.  Doing any other kind of experiment, it is trivial to tell the difference.  Dropping the masses, not so much.  I do not claim that wherever one sees an inertial mass m, freely drop in a Planck mass, that would be silly.  This blog is highly restricted to dealing with gravity.

In the suggested upgrade to the universal gravity law, there is a Planck mass on only one side of the equation, the one that is a product with acceleration.  A Planck mass is a name for a particular set of constants that dimensionally have the units of mass.  On the potential side of the equations there is a different collection of constants and it does not have the dimensions of mass (distance cubed per second squared, probably no nice label for that).

The deepest quality of using the collection of constants known as a Planck mass is that a Planck mass does not attract.  They are just constants, nothing more, nothing less.  No one can point to the Planck mass.  Experiments can be done to measure the three different constants involved (or as in the case of the speed of light, the speed could be defined).

In a different proposal, you suggest one could use an arbitrary mass that you arbitrarily set to one kilogram.  To use that law, one would need to reference the one kilogram in Paris.  The experimentalists also need to measure Newton's gravitational constant G.  This is a subtle simplification that my upgrade provides: only experimentalist are needed to calculate the constants instead of those same experimentalists plus a standards committee.

Gravity is not about inertial mass, it is all about how gravitational masses change space-time geometry.  I thought only energy that contributes to the stress energy tensor is the cause for the pure geometry change that is the Ricci tensor stuff on the other side of the Einstein field equations.  The upgrade in this blog is an effort toward be consistent with that interpretation of the Einstein field equations.

> But the "c" parameter which was sent off to infinity there is no longer connected with the speed of light.

This is the kind of issue I trip over.  I should have stuck with the instantaneous propagation of change issue in the gravitational field equations.  Maxwell remains Maxwell, and those equations say the speed of light is finite.  The problem is about gravity, not light.  I stand corrected on this point.

I am a bit unsure of how to respond, since you are (not consciously of course) contradicting yourself or squinting to see things a certain way, and may just need time for things to settle in place.

If this message doesn't help, maybe just sit on it a few days and try again.  I find that helps me sometimes when trying to step back and see something clearly.
Is there any experiment in classical mechanics that allows someone to tell the difference between the inertial masses of the spoon, the moon, and a photon?  It may well be that every such experiment allows one to tell the difference.  I can toss a spoon against a wall.  I cannot change the motion of the Moon by pushing against it since it has far too large an inertial mass.
Exactly, as soon as you use a force other than gravity (ie. you pushing an object in that example), the inertial mass is apparent in Newtonian mechanics.  So the inertial mass does matter.
This blog was not about inertial masses in classical mechanics.  This blog was exclusively about the fundamental force of gravity.
As I explained above, your starting point had more than just the force of gravity in it.  You included Newtonian mechanics, and included the force of gravity, you got an equation of motion for a free falling mass, and then changed both sides of that equation.

Let me try to explain again. In Newtonian mechanics we have:

To get an equation of motion for an object, we then need to specify all the forces acting on that object, and calculate the net force.  That net force can then be put in this equation to determine the motion.
Newton's law of gravity says the force between two gravitational masses m1 and m2 is:

If the only force on a particle is gravity, then Fnet = Fgrav.  If the gravitational force is due to only a single mass M, then we find a particle with m_grav and m_inertial moves according to:

You started with that equation of motion.  And you changed both the m_grav and m_inertial to your universal mass to get:

which is your new law -- your new equation of motion in gravity.

I do not claim that wherever one sees an inertial mass m, freely drop in a Planck mass, that would be silly.  This blog is highly restricted to dealing with gravity.
I realize you didn't say that explicitly.  The point is that you didn't restrict yourself to changing the Newtonian law of gravity.  If you merely did that, you've have:

and leaving Newtonian mechanics alone...

would then give the equation of motion as

which is not what you proposed.  You changed both Newtonian mechanics and the force law.

Also, writing the force law explicitly like that

helps make it more clear why I asked "How does the universe decide which masses to ignore in this idea?".  Why replace m_1 with a universal mass constant, but not m_2 ?

I basically have just repeated my same points here, so if my objections still don't make sense, let's just let it simmer for awhile.  I think if you come back and see it fresh in a couple days, it will be so obvious to you, that you'll wonder how you didn't see it before.
Let me see if I can state why you think there is a logical problem (an illegal substitution).  I will take a longer road since it may clarify other issues along the way.  I describe the weak and strong equivalence principles.  Let me begin by starting with a bit of classical physics, the simple harmonic oscillator.
Simple harmonic oscillator

Here is the force equation for a simple harmonic motion:

$m_i \frac{d^2 x}{dt} = - k x$

I am using a subscript i to say this mass in an inertial mass.   Double the inertial mass, and the force equation changes.  This is the domain of classical mechanics.  I had better not change this, or the proposal is silly (a nice way of saying it is so wrong it is not worth considering).

The weak equivalence principle

The weak equivalence principle can be understood by looking at Newton's gravitational force law, this time with three subscripts: i for inertial mass, p for passive gravitational mass, and a for active gravitational mass.

$m_i \frac{d^2 x}{dt} = - G ~\frac{m_p M_a}{R^2}$

The weak equivalence principle (wep) is the claim that the inertial mass is precisely equal to the passive gravitational mass.  The modern tests of the weak equivalence principle started with Eötvös.  All experiments of this type have shown the same thing: to the limits of the experimental tests, the inertial gravitational mass is exactly equal to the passive gravitational mass.

Newton's law of gravity - as Newton wrote it - is consistent with the weak equivalence principle.  In Newton's law, there is both a inertial mass and a passive gravitational mass.  It is an assumption that both are exactly the same, an assumption backed up by experimental data.

In the upgraded Newton's law, there are constants standing in both slots.  As such, without doing an experiment, the upgrade is consistent with the weak equivalence principle: the same set of constants are equal to each other.

The strong equivalence principle

This is a different idea.  The idea is that gravity is exclusively about a dynamic space-time geometry.  There can be one and only one dynamic metric that describes everything that happens due to gravity.  The tests for this are energy loss by binary pulsars.  For more complicated models of gravity, there will be a dipole mode of gravitational wave emission.  For models that are consistent with the strong equivalence principle, the lowest mode of gravity wave emission is the quadrapole.  Newton's law of gravity can be written entirely as a metric theory of Galilean space-time.  As such, Newton's good old law is consistent with the strong equivalence principle.  The gravitational wave will not be the same strength as predicted by general relativity, but the lowest mode of emission for an isolated source is the quadrapole.

Newtonian gravity and special relativity

Every test of special relativity indicates Newton's law of gravity must be wrong.  The upgrade I suggest in no way alters this fatal flaw.  Galilean geometry is not right, Lorentz is.

The illegal substitution

Let's write out two force laws, one for a simple harmonic oscillator (say a slinky), and another for gravity (with the very same slinky):

\begin{align*} m_i \frac{d^2 x}{dt} &= -k x\\ m_i \frac{d^2 x}{dt} &= - G ~\frac{m_p M_a}{R^2} \end{align*}

Forces add up, so we can simplify this:

$m_i \frac{d^2 x}{dt} &= -k x - G ~\frac{m_p M_a}{R^2}$

Now do the substitution suggested in this blog:

$\sqrt{\frac{c \hbar}{G}} \frac{d^2 x}{dt} &= -k x - \sqrt{G c \hbar} ~\frac{M_a}{R^2}$

That sure looks wrong to me.  Silly, as in really, really wrong.  What I have claimed is that the substitution can only be done on a force law where one has an inertial mass on one side and a passive gravitational mass on the other.

\begin{align*} m_i \frac{d^2 x}{dt} &= -k x\\ \sqrt{\frac{c \hbar}{G}} \frac{d^2 x}{dt} &= - \sqrt{G c \hbar} ~\frac{M_a}{R^2} \end{align*}

I will spend a few days thinking about this pair of equations.  I can see how if you wished to preserve the superposition of forces law, this proposal is silly.  That assumes one should treat gravity like any other force law.

Gravity is not like any other force law.  Gravity is about a dynamic changes to [Galilean/Lorentz] space-time geometry.  Do an experiment where there is a gravity field, then every other force law, even the simple harmonic oscillator, is changed in a known way by gravity.

Anyway, that is something worth thinking about: superposition of force laws versus dynamic [Galilean/Lorentz] space-time metric laws.
Anyway, that is something worth thinking about: superposition of force laws versus dynamic [Galilean/Lorentz] space-time metric laws.
Newtonian mechanics actually has a nice way of looking at this.  If you go to a non-inertial frame, for example a rotating frame, a pseudo-force will show up that is proportional to the inertial mass.  In this sense there is a hint that a free falling frame is actually a (local) inertial frame; gravity is some how just a pseudo-force.

So for now, to keep everything on equal footing for comparison, just stick to normal flat geometry, and describe all the effects with forces.
Let's write out two force laws, one for a simple harmonic oscillator (say a slinky), and another for gravity (with the very same slinky):

\begin{align*} m_i \frac{d^2 x}{dt} &= -k x\\ m_i \frac{d^2 x}{dt} &= - G ~\frac{m_p M_a}{R^2} \end{align*}

Forces add up, so we can simplify this:
$m_i \frac{d^2 x}{dt} &= -k x - G ~\frac{m_p M_a}{R^2}$
We seem to be on the same page now, so this may just be a silly terminology issue, but I'm still worried about you calling those force laws.  Those are equations of motion, which are obtained by taking Fnet = ma, and the net force is calculated by including various force laws.  For example Hooke's law for the spring, or Newton's law of gravity.

You clearly understand this, since when you include a spring in a gravitational field, you don't add those two equations together to get:

So be more explicit with that step.  You have from Newton's laws of motion (Newtonian Mechanics):

Then we have various definitions for forces (Hooke's law, and Newton's gravity):

Given a situation, we
1) calculate each force acting on the object
2) add the forces to get Fnet
3) put the Fnet result into Newton's second law of motion to obtain an equation of motion

You think you are only manipulating Newton's law of gravity, because you keep calling an equation of motion in gravity the law of gravity.  But in reality you are manipulating everything that went into calculating that equation of motion, including Newtonian mechanics.

As another example, imagine the forces on an object are gravitational, but due to the Moon and the Earth or any number of massive objects.  How does your "force law" change?  What steps do you take?  It seems to me you will find when being explicit with the steps that ...

You are actually trying to say the gravitational force from a mass M_a is:

Then we add those forces from each gravitating body to obtain Fnet.  Then we put Fnet into Newtonian mechanics (or your modification), to get the equation of motion.  To get your equation of motion your replacement for Newtonian mechanics is:

So while I know what you mean when you call an entire equation of motion the 'law of gravity', this slip in terminology I think accidentally led you to believe you could adjust the whole equation of motion and only be suggesting a new 'law of gravity'.  Instead you accidentally changed both gravity and newtonian mechanics.
Thanks for your critiques.  I have added a line at the top of the blog because I think I must write down the action for this proposal.  In a future blog, I will have an action where one can see an illegal substitution that causes the change in Newtonian mechanics (just to save you from doing the work).  There may be a different action which makes sense - or it might not - but that will require some effort (after the holiday).  I will leave this blog as is because it was the way I initially thought of the idea: just substitute in the Planck mass only into the gravitational force equation and go.  I did not think of the consequences you pointed out, so thanks.

Another reason for a new blog: I will seek to curtail people from proposing their own, quite different proposals.

Tomorrow will be the holiday card blog :-)
Doug: I think you'd be better off thinking about "space-time metric laws" all on their own. IMHO you should start on why the path of a photon moving through space is curved. Remember what Newton said: ...grow denser and denser by degrees... After that I think you should move on to a single electron. I say this because there is no external force acting upon a falling brick. The principle of equivalence relates the force on a brick on the ground to the force on an accelerating brick in free space. But the brick on the ground doesn't move, so no energy is added until you lift it. Then you do work on it. Then force x distance = energy applies. Then you've added energy. But when you drop it, PE is converted into KE, that's all. No more energy is added. Hence when you drop a 1kg brick into a black hole, the black hole mass increases by 1kg.
I used to call myself the "ultra-conservative fringe".  Since writing that blog, my proposal using quaternions and hypercomplex numbers was shown to be 100% wrong by Henry and LagrangiansForBreakfast.  That experience drove me to a new label, "ultra-orthodox fringe" where my only hope in making a contribution lies in pointing out assumptions made in standard approaches.  The tool set I use though must be conventional.  As such, I will be doing the conventional analysis by starting with actions.

Please do not take this personally as I bet you are a nice person to share a gluten-free beer with.  Since both Henry and LFB are on your case for statements made here, I don't trust your advice at all.  Those were the same guys who took down a proposal I had talked about for years, and I have \$800 worth of t-shirts with bad math to prove it.  I will trust their assessments before yours.

In my future blogs, I will keep you on a short leash.  You do have plenty of ideas about gravity, so write a blog about it.  In the future, I will not let my blogs be your platform.  Just a heads up.
"In the future, I will not let my blogs be your platform. Just a heads up."

For every reader out here, let me say thank you Doug.

John Duffield is actually a crackpot looking for a platform.
He is also known as "Farsight" -- a forum moderator on anti-relativity.com
He even promotes a "theory of everything" in the most crackpot sense, with no useful math descriptions.
It is impossible to claim you understand mainstream physics when you give an interview like this:
http://www.richplanet.net/starship_main.php?ref=7&part=1

He's not a troll, but feeding a crackpot who wants a platform is actually way worse.

So thank you Doug. Thank you.

Doug: do start with actions, the h in E=hf is Planck's constant of action. And the massless photon has its gravitational field.

Henry and LFB are on my case about invariant mass. I've supported this with references which they've ignored, and they're casting ad-hominem aspersions at me whilst evading the issue: if you drop a 1kg brick into a black hole, the black hole mass increases by 1kg. So where did the kinetic energy come from? From the black hole via some action-at-distance magic? No. From the gravitational field when there is no actual force acting on the brick? No. That's the whole point about relativity, the "force" of gravity is NOT a force in the Newtonian sense. The falling brick is not gaining energy. Conservation of energy applies. The kinetic energy comes from the brick. Potential energy in the brick is converted into the brick's kinetic energy. There is nothing unorthodox about that. It is utterly conventional.

So if you get rid of that kinetic energy, what's happened to the rest mass? Press them on this point. Your trusted advisors are trying to deflect you from this. They are being evasive. And they've shot down everything you've tried. Have a think about that. Think about who's really trying to help you here.
"The upgrade make's Newton's law more universal because it can also be used to describe the motion of light."

You have not upgraded the law; you just applied it to the Planck mass. However, at that level the gravitational force is negligible and there are other forces that dominate. In addition, there is no determinism but uncertainty and Newton's 2nd law does not apply since position and momentum cannot be known simultaneously.

There are many issues with Newton's 2nd law. One major issue is that it cannot even be derived from Newton's Principia unless many assumptions are made that are not evident. Newton just said that the change in some quantity of motion (thought to be momentum) is propositional to some force. This does not directly equate to F = dp/dt unless one goes too far to assume it. Euler many years after Newton proposed the equation F =ma as a better expression of the law when the inertial mass stays constant.

I have proposed an alternative systems of laws of motion based on power and Newton's law appears to be a subset of these because they allow inertial curvilinear motion. See: http://www.digitalcosmology.com/

Interesting blog Doug. As Edward says, you left the big M in there. But I think you're right to be thinking along these lines, and thinking for yourself. I would say however that you're getting a bit bogged down with mass here. Take a look at Einstein's E=mc² paper. Look at the very end, where he says radiation conveys inertia between the emitting and absorbing bodies. Now read this article. Light has a zero rest mass. But its inertial mass is NOT zero. You can't slow down a photon like you can slow down a spoon. But you can decelerate it. That's what Compton scattering does. The Inverse Compton accelerates it. This is an acceleration in the vector sense, but it's still an acceleration. The photon's inertial mass is the same as its active gravitational mass. The photon causes gravity just like any other concentration of energy. And the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy. That's on page 185 of the Doc 30 Foundation of the General Theory of Relativity. It's energy that matters, not mass. Because of the wave nature of matter. Because photon momentum is resistance to change-in-motion for a wave propagating linearly at c, whilst electron mass is resistance to change-in-motion for a wave going round and round at c.
> Light has a zero rest mass.

We all of course know what you mean, but that phrase makes me cringe a bit.  Just say "invariant mass", or "proper mass" if you want to distinguish that definition of mass.

> But its inertial mass is NOT zero.

It's a bit of a stretch, but I believe I know what you mean here as well.  While light has no proper mass, it still has momentum.  This momentum can change in Compton scattering, or reflection, and impart an impulse in doing so.  I do not recommend referring to this as inertial mass though.

> The photon's inertial mass is the same as its active gravitational mass.  The photon causes gravity just like any other concentration of energy.

Now this is stretching to the point of being, at the very least, misleading.
You seem to be suggesting something along the lines of inertial mass = gravitational mass = relativistic mass = E/c^2.  Then for a photon moving in the gravitational field of a massive particle M we get:

So now gravity can deflect light, and light can also contribute to a gravitational force.

This makes me uncomfortable for multiple reason, the first being the usual obvious objections to equating inertial mass and relativistic mass.  So this really is misleading.  The whole issue of throwing a quantum object (a photon instead of maxwell's equations) into a classical theory, well I'll just avoid that topic for now.

> The photon causes gravity just like any other concentration of energy.

Since you refer to GR, this could also be taken in the context of GR.  If so, you seem to be implying that in GR the source of gravity is the energy density. That is not the case.

The gravitational source in GR is the stress energy tensor, and electromagnetic radiation does contribute to this tensor.  But it is a tensor and doesn't contribute as some single number "mass" though.  There is no scalar "gravitational mass" in GR like there is in Newtonian mechanics.

A concentration of energy in photons will NOT necessarily cause gravity just like the same concentration of energy from some massive field.  The energy density is only a single component of the stress energy tensor.

> And the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy.

This is incredibly misleading out of context.  Consider a gravitational wave.  It does not contribute to the stress energy tensor.  Or instead of a wave, consider the static curvature in space outside a massive body, this also does not contribute to the stress energy tensor.

Based on your previous posts, you seem to take layman's descriptions too seriously to the point of misunderstanding GR severely.  If we merely have different preferred terminology, that can affect communication, but is really no big deal.  However if there is actual differences in understanding the physics predictions, that is a big deal.  So I want to make it very clear up front that these statements you are making can be misleading or wrong if taken too seriously.
Edward: l used the term "rest mass" because invariant mass is a misnomer. The "invariant" mass of a brick increases when you lift it. I take your point on inertial mass, I was rather following the convention wherein "active gravitational mass" is equivalent to "inertial mass". As it happens I don't like to see the word "mass" used for anything other than rest mass. But remember that radiation conveys inertia between the emitting and absorbing bodies, and that you can indeed accelerate a photon.

I think you've misread what I'm saying. The photon could be moving through space all on its own. It causes gravity. It's so slight for a single photon that we cannot hope to measure it. But we know that the inertia of a body is a measure of its energy content, and that we can trap a photon in a mirror box. Then we can easily conceive of a gedanken mirror-box containing a very large number of photons. Every time we trap a further photon in the mirror-box the mass of the system increases, as does the resultant gravitational field. This doesn't disappear when we open the box.

No problem re "gravitational mass", see above. The point I was making is that mass is a red herring when it comes to gravity. It doesn't matter whether you have a 511keV photon or a 511keV electron, you still have a gravitational field. But please do note that the stress-energy-momentum tensor is something of an artefact. It's essentially a matrix describing energy-momentum-pressure etc, and you need to have some underlying field/particle/wave to provide this. The important point is that if you have some concentration of energy, you will have a gravitational field. You cannot have a situation where you have an energy density of x in some region and less than x in the surrounding region, and no gravitational field.

I can see no way in which this is misleading: the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy. Any concentration of energy in any guise will result in a gravitational field. With respect, what is misleading is the static curvature in space outside a massive body. Curved spacetime is not curved space.

Based on my previous posts I hope you appreciate that I take Einstein's descriptions seriously. I don't think there's any difference re the physics predictions. But I do think there's a difference in understanding the physics, and that this is very important.
I'm not sure how to say this politely, so I will be straight forward: Are you capable of handling the math of arbitrary coordinate systems in SR or calculating things given a metric solution in GR?  I am trying hard to assume good faith, but it very much appears that you have a very poor understanding of this material which goes no deeper than a collection of ill fitting layman's explanations which you misapply.

So let's find some common ground by avoiding any layman descriptions and discuss some math.  First some notation to get us started on the same page.
•   Let's not carry around factors of c or G, as those are clear from context so we'll just choose units such that they are 1.
•   Let's also use the standard Einstein tensor notation with implied summation rules to simplify expressions.
•   For coordinate and sign convention, in inertial coordinate systems the order of component will be (t,x,y,z) and the diagonal of the metric is (1,-1,-1,-1).
•   Energy and momentum of a particle will be shorthand for referring to the components of the contravariant four-momentum in some coordinate representation as such:

(and like most textbooks, \mathbf will signify a three-vector as was done with there).
• For our purposes here let's define the proper mass of an object as the positive invariant obtained from the four-momentum:

In inertial coordinates this gives the usual:
I think that's all the notation issues that may come up here.  Feel free to define others you need as they come up. I'd like to stick to the conventions in MTW for convenience if possible (You have that book I assume?  Almost everyone who's studied GR seems to end up with it.  If not, what textbook do you turn to for reference?)

Alright, let's start with an easy one:
l used the term "rest mass" because invariant mass is a misnomer. The "invariant" mass of a brick increases when you lift it.
This unfortunately is wrong.  To avoid the messiness of including other forces, let's just consider a free-fall situation.  From above, you appear to be claiming the invariant mass of a particle free falling into a black hole will change over time.  Is that a correct statement based on your understanding?

So now, please try to work out the math to back up your statement.  If this does not help you find the flaw in your understanding, it will at least allow us to see where your math mistakes are so that we can help.

The GR exercise:
Consider a massive point particle moving in the curved spacetime given by the Schwarzschild solution.  Use whatever coordinate system you are comfortable with for the calculation, and calculate the proper mass as a function of time (coordinate time or proper time, whatever you choose) as the particle free falls when released from rest at t=0 (from some position of your choosing).

Please show your math.  It is time to step out from behind the shield of layman's explanation and help you fix your understanding of relativity.

With respect, what is misleading is "the static curvature in space outside a massive body". Curved spacetime is not curved space.
Noted. I see that you are sensitive to that shorthand and will try to explicitly say "spacetime" instead.

You however seem to have ignored my actual objection (gravity waves do NOT contribute energy to the stress energy tensor), and similarly just repeated your misunderstandings for other topics as well.  So let's just focus on the huge red flag of "invariant mass is a misnomer" issue first.
Henry: I can handle the maths of SR, I can plod slowly through the maths of GR.

Re notation, I'm afraid it is not clear that c is 1. The coordinate speed of light varies with gravitational potential. And you will be aware of the Einstein quotes saying the speed of light varies with position. This causes issues for four-momentum.

I do not have a copy of MTW, and refer to no set textbook. However I do refer to Einstein, and I do think for myself.

Re the mass of the brick: yes, the “invariant” mass of a brick increases when you lift it. Yes, the “invariant” mass of a particle or brick falling into a black hole decreases over time. Yes, this is a correct statement of my understanding. Please refer to mass in general relativity on Wikipedia and note this line: Non-covariance of the energy-momentum four-vector implies non-invariance of its length, the invariant mass.

It isn’t maths that backs up my statement. Conservation of energy backs up my statement. Throw a 1kg brick into a black hole and the black hole mass increases by 1kg. There is no flaw in my understanding of that. And you cannot dispute it.

”The GR exercise: Consider a massive point particle moving in the curved spacetime given by the Schwarzschild solution. Use whatever coordinate system you are comfortable with for the calculation, and calculate the proper mass as a function of time (coordinate time or proper time, whatever you choose) as the particle free falls when released from rest at t=0 (from some position of your choosing).”

Responding to this would be very time consuming for me, so I will politely decline. But please see http://arxiv.org/abs/1212.0263 and note this ” the energy used to lift a body in a static gravitational field increases its rest mass”. My understanding might be unfamiliar to you, but it is not something unique to me.

Yes, please do say explicitly say spacetime when you mean it. This is very important.

I didn’t deliberately ignore your objection concerning gravitational waves.

Yes, let’s focus on "invariant mass is a misnomer".

"What do you think happens?"

Invariant mass varies. That’s why we have the mass defect. Our 1kg brick ended up moving at a relativistic speed. And yet the black hole mass increased by only 1kg. I'm afraid something has got to give, Henry. And that thing is invariant mass. Once you have appreciated this, I'm afraid it gets worse.
It has long become clear we need to get to explicit math to come to an agreement, so let's not give up on that so quickly.
Re notation, I'm afraid it is not clear that c is 1.
I don't want to get into this argument now (especially since you refused to specify the math to help us relate your aether parameter idea of relativity to the mainstream version).  So instead, I will point out that you already said the mainstream way of calculating will not give different predictions.  Therefore I suggest for the notation and calculations, we stick to mainstream physics.  Since the predictions will not be affected, the main change will be that we can communicate easier, which is a good thing.  Agreed?
It isn’t maths that backs up my statement. Conservation of energy backs up my statement. Throw a 1kg brick into a black hole and the black hole mass increases by 1kg. There is no flaw in my understanding of that. And you cannot dispute it.
There is a flaw if you think this supports your claim that the invariant mass of a particle changes as it falls into a blackhole.  The wiki article you link doesn't even support your claims (and ironically is talking about problems of discussing conservation of energy in GR, the exact opposite to one of your statements here).  I can't be sure what the heck the flaw in your logic is, because it is not clear to any of us what "logic" you are even using to connect those statements to your claims.  You continue to just keep making disconnected confusing statements like above instead of showing the math to make it clear how you think those statements are logically connected to your claims on invariant mass.  We need to cut through the ambiguity of disagreements on terminology; we need to see the math.
Responding to this would be very time consuming for me, so I will politely decline.
Please take the time.  There is a reason why Doug's crackpot rule is to require people that are making strange physics statements to specify it in math. We will waste much more time going in circles without math since we can't even agree on the meaning of some words.

So please, take the time to explain your statement in math by working out that GR exercise.

Let me give some nudges in a direction.  I will help you out, but I will not do the math for you, since the point of this exercise is to let us see how your understanding translates into the math. To see "behind the curtain" as LFB put it.  OK?  So here's a question to think on to help you out:  Do you agree that a free particle moving in curved spacetime has, by definition of being a free particle, zero four-force acting on it, and a zero proper-acceleration?
"Do you agree that a free particle moving in curved spacetime has, by definition of being a free particle, zero four-force acting on it, and a zero proper-acceleration?"

Yes, see my comment to Doug of 12/23/13 11:54. I said there is no force acting upon the falling brick.

Now, do you agree that when a 1kg brick falls into a black hole, the black hole mass increases by 1kg?

Do not try to cast aspersions or create some distraction. Just answer the question.

LFB: you started this. You answer that question too. And then you can answer the next question.

Doug: pay attention to this. It will be revealing.
In that comment you referenced you wrote:
I've supported this with references which they've ignored, and they're casting ad-hominem aspersions at me whilst evading the issue: if you drop a 1kg brick into a black hole, the black hole mass increases by 1kg. So where did the kinetic energy come from? From the black hole via some action-at-distance magic? No. From the gravitational field when there is no actual force acting on the brick? No.
I did not ignore your links, and neither did LFB.  The references do not support your statements, and I have pointed this out as has LFB.  It is clear you don't understand enough of the math and physics to pull relevant arguments from an article.  And this is not an ad-hominem attack.  Ad-hominem is rejecting a claim on the basis of some irrelevant fact about the author of or the person presenting the claim or argument.  Your lack of math and physics skills is not only extremely pertinent to the issues we're discussing, but are actually the cause of your incorrect claims.

In that above quote you seem to think kinetic energy shows up at the expense of the invariant mass. This is an example of why I'd like to see your math.  This is so far from a correct understanding of GR, that we really need to get beyond your terminology issues and see the unarguable result of the math.  If you'd like me to try to explain without math, here you go: kinetic energy is just a coordinate dependent effect.  Go into the instantaneous inertial rest frame of the particle and there is no kinetic energy.  So choose some coordinate system, and yes you can see the kinetic energy and momentum change as the particle falls, but the invariant mass of the particle stays the same.

If you feel we are just misunderstanding you, then please respond to my GR exercise requesting you show the actual math. In trying to work that out, I believe you are smart enough to find your own errors.  If not, we will be able to help you more directly.
Now, do you agree that when a 1kg brick falls into a black hole, the black hole mass increases by 1kg?

Do not try to cast aspersions or create some distraction. Just answer the question.
As LFB mentioned, invariant mass doesn't just simply add.  But ignoring details from that and minute energy lost during ring down, etc., yes the black hole mass increases by 1kg.

Again this doesn't support your claim that a massive particle will lose invariant mass while free falling into a black hole.

In response to my question:
"Do you agree that a free particle moving in curved spacetime has, by definition of being a free particle, zero four-force acting on it, and a zero proper-acceleration?"

You wrote:
Yes, see my comment to Doug of 12/23/13 11:54. I said there is no force acting upon the falling brick.
So you are almost there.  If you understand that the four-force is zero, it should not be that hard to choose a coordinate system now and to calculate the four-momentum as the particle falls into the blackhole.

Use whatever coordinate system you want, but here's another hint: Do you agree that locally in a tiny free falling lab, experiments will show this to be like moving in flat-spacetime; ie. we can choose a coordinate system such that the coordinate components of the metric in a free-falling frame is locally equivalent to that in an inertial frame?

You are almost there.  This does not need to be an incredibly involved calculation.  Use the freedom to choose a coordinate system to your advantage to help simplify the math.  If you need to see it in multiple coordinate systems to convince yourself, go for it, but that will be much more involved.

So please stop avoiding the request to show math to back up your claims.

----------------------------------
The GR exercise:
Consider a massive point particle moving in the curved spacetime given by the Schwarzschild solution.  Use whatever coordinate system you are comfortable with for the calculation, and calculate the proper mass as a function of time (coordinate time or proper time, whatever you choose) as the particle free falls when released from rest at t=0 (from some position of your choosing).

Please show your math.  It is time to step out from behind the shield of layman's explanation and help you fix your understanding of relativity.
Let's cut right through the huff and puff, Henry. Let's cut to the chase. Your words:

"yes the black hole mass increases by 1kg".

OK, let us zoom down in our gedanken spaceship and intercept that 1kg falling brick somewhere comfortably above the event horizon. Let us slow that brick down to a halt, getting rid of its kinetic energy by radiating that kinetic energy into space.

Now, after doing that, let us once more drop that brick into the black hole. Then we can zoom back to where we were when we dropped it the first time.

Does the black hole mass increase by 1kg Henry?

You know the answer is No, not now it doesn't. Because you know about conservation of energy. Don't you Henry? So you know that I am right.

Does anybody else want to dispute this? Because if you do, I'm throwing down the gauntlet. Because Doug has got to understand these simple little things before he can make progress. He's got to understand that gravity is the reaction to the action, because if he doesn't, he's going to be shot down forever by people who are just getting in the way.
And again you don't show your math.

John,
What will it take for you to actually work out that exercise and show the math backing up your claim?

Stop avoiding the request for you to back up your claims with math.  You say you understand the theory, and that your interpretation is equivalent to mainstream physics at least in prediction for comparison to experiment.  But your understanding appears just blatantly wrong.  Why are you unwilling to actually show the math?  I have not even asked for a particularly difficult exercise.
Let us slow that brick down to a halt,
At rest according to what coordinate system?  "to a halt" is a coordinate system dependent statement.  At any point along its path the brick can already be found to be at rest according to many different coordinate systems.  So I'll choose one, and "release" it from there.  Of course nothing has changed.  So of course the blackhole mass increases the same amount.

This is the problem with your laymen descriptions. They are imprecise, and would only require more questions to get to what you mean.  But due to your misunderstanding and terminology issues, I have become confident this will just lead further astray.  So stop this.  Please just actually calculate the math to support your statement.
So you know that I am right.

Does anybody else want to dispute this? Because if you do, I'm throwing down the gauntlet.
Does "throwing down the gauntlet" mean you will actually show math to back up your statements?  Because I dispute that you are right.

So if you are so confident you are correct, please stop avoiding the math exercise and show your work.
(Please also respond at the bottom of the comments to reset the indenting.)
I have waited, and you still have not shown any math to back up your claims.  Since you claim you can handle the math of GR, and we both have accounts on this site, I propose the following:
We both work out the exercise given previously, and post our answers showing the math as blog entries on this site.  I propose Saturday 1/11/2014 at 6pm GMT as a deadline, which would give us more than a week to find the time to write up the solution to the exercise.  Do you agree to this proposal?

This way both of us will show the math for that exercise and hopefully get to the root of the physics disagreement.  Please tell me if you agree to this.  Because I am not going to waste my time writing up the math and an explanation if you aren't ever willing to go past your talking points and actually work out the math of that exercise.
John Duffield: "I do not have a copy of MTW, and refer to no set textbook."

Let me be more blunt than Edward. Have you taken the time to learn GR from ANY college textbook?
And have you actually worked through the math of homework problems in the book to learn the material?

In your profile, you mention that this isn't even your field. Why is it so hard for you to at least be humble enough to consider you don't understand the material?

"Let me be more blunt than Edward. Have you taken the time to learn GR from ANY college textbook?"

No. I've taken the time to learn GR from the original material by Einstein. I recommend it. And let me be blunt. When you do this, you find it squares with the hard-scientific evidence. But then you come to appreciate that some things in some college textbooks don't square with the original material and the hard scientific evidence. You come to appreciate that some things in some college textbooks are wrong. And yet people treat their textbook like a bible, so much so that they actually dismiss Einstein and the hard scientific evidence. When you challenge them on this with a simple scenario and concise clear logic with robust references, you don't get the same in return. You get abuse and evasion instead, because they brook no challenge, and they are not sincere.

By the way, somebody mentioned the anti-relativity forum where I was a moderator. If you dig around a little you'll find that I was a moderator on that forum despite being pro-relativity. Because I am civil, and I do not resort to ad-hominems in order to evade a physics point. A telling point: when you drop a 1kg brick into a black hole, the mass of the black hole increases by 1kg. Would you like to have a go at that one? Or are you some textbook author looking forward to continuing royalties and fighting shy of that famous comment by Max Planck? You know, Max Planck? The guy who published Einstein's papers? The guy who said science advances one funeral at a time. This hasn't quite clicked with Doug yet. But it will.
"college textbooks don't square with the original material and the hard scientific evidence."

"they actually dismiss Einstein and the hard scientific evidence."

You've used that phrase many times. What exactly do you mean by "hard scientific evidence"? When I hear that, I think of the prediction of a theory compared to experiment. Are you saying that the theory presented in college textbooks is not equivalent to Einstein's, and that experiments show Einstein's theory is correct while college textbook's theory is wrong?

That "light is heavy" page is making a lot of mistakes, or borrowing Henry's polite wording, it is "at the very least, misleading". Note to readers, the "G. 't Hooft" author is not the famous one. Not that it matters who said wrong statements, but I just wanted to quickly dispel any possible misunderstanding there.

When searching about that paper you linked, I hit upon your discussion earlier this year here:
http://backreaction.blogspot.com/2013/07/how-stable-is-photon-yes-photon...
There you make similar bizarre statements. Including your belief that the invariant mass of a particle somehow varies in GR. What the heck!? Your understanding of this material is very very poor. It appears your main obstacle to actually learning, is an inability to accept that you may be wrong and an inability to actually check the math yourself to find out. Is there anything any of us can say that will snap you out of this?

You cannot get a reasonable understanding of a theory by collecting a bunch of layman's explanations of different features of the theory. There is no replacement to actually learning the math.

LFB: Apologies, I've referred to "Light is Heavy" before and made it clear that the 't Hooft isn't the Nobel Laureate. I forgot to do it this time. Where is this article wrong by the way?

Yes, the mass of a particle varies. When you lift a brick you do work on it. Conservation of energy applies. The mass of the brick is now increased. When you drop the brick the "potential energy" within the brick is converted into kinetic energy. Once this is dissipated the mass of the brick is back to its original lower value. Yes one can talk about a system, but when you launch the brick upwards at 11km/s, conservation of p = mv momentum and kinetic energy = ½mv² tells you that the brick gets the lion's share of the kinetic energy. It departs the system taking that energy away with it. This is how it is, LFB. It's the same for an electron. Have a look at "mass deficit".

"Your understanding of this material is very very poor. It appears your main obstacle to actually learning, is an inability to accept that you may be wrong and an inability to actually check the math yourself to find out. Is there anything any of us can say that will snap you out of this?"

With respect it isn't very poor. It's very good. Now please go and look it up. Read about the mass defect and binding energy. The mass of a hydrogen atom is less than the mass of a proton plus the mass of an electron. When you have looked it up, you might like to apologise for your "learn the math" admonishment. I have learned the math. And moreover I have understood the terms. That's why I can talk confidently about terms like m.
> Where is this article wrong by the way?

If you actually tried to learn from what people have been posting, you'd already know the answer. Henry in particular explained (quite clearly in my opinion) many issues with your understanding that also apply directly to that paper.

> With respect it isn't very poor. It's very good. ... I have learned the math.
> And moreover I have understood the terms. That's why I can talk confidently about terms like m.

It is terrifying how far your confidence is from your actual level of understanding.
Are you even aware that you are disagreeing with mainstream physics, or are you under the false impression that you are defending it?
What will it take for you to seriously consider that you are wrong?

You seem to be under the impression that you understand SR and GR fine, but you just prefer different terminology and interpretation. But what you are saying doesn't even appear to be self-consistent, and other things seem to strain all credulity to claim a terminology difference could fix it. And all my requests (and requests from others as well) for you to clearly explain how your "interpretation" and "terminology" maps onto math, well they have been ignored so far. It's like you are using claims of terminology differences as a thick curtain to claim that we can't see behind the curtain, but you're back there and we just need to trust you understand it somehow. Please, no more hiding behind the curtain. I'd appreciate it if you want back to our discussion on Doug's previous blog post and respond to the requests to define the details of your aether parameter / "conditioned space" view that you think is not a layman's explanation or approximation but actually equivalent to GR. (There's also some good Einstein quotes put there recently showing you are cherry picking when arguing by authority as well.)

It appears the only way forward then is to look at the math, which will hopefully strip away any terminology issues and get to the core problem. Please do respond to our questions requesting precise math. Currently only you can see 'behind your curtain', so give us the math details so we can check if your weird interpretations are actually mathematically equivalent to mainstream physics.

LFB: I asked you to say why the article was wrong, and you have not.

I am not disagreeing with mainstream physics when I refer to the mass defect and point out that "invariant" mass varies. Please refer to mass in general relativity. When you throw a 1kg brick into a black hole, the mass of the black hole increases by 1kg. On the way in, that brick has considerable kinetic energy, but at all times its rest mass plus kinetic energy, aka "relativistic mass" has a mass-equivalence of 1kg. The mass of the black hole does not increase by 2kg, or 3kg, or 4kg. It increases by 1kg. Conservation of energy applies. Please concede this simple point instead of retreating behind ad-hominems and mathematical demands. For it is crucial to the distinction between Newtonian gravity and Relativistic gravity, and therefore crucial to this discussion. The "force" of gravity does not add energy to our falling brick, it merely converts potential energy within the brick into the brick's kinetic energy.
> I asked you to say why the article was wrong, and you have not.

Do you want me to just copy paste Henry's statements?
Here's an easy one, it equates inertial mass with relativistic mass. It also tries to claim relativistic mass is the source of gravity.

Relativistic mass is not even a very useful concept. Trying to work with relativistic mass, you come to realize you actually need a direction dependent mass (a "longitudinal mass" and a "transverse mass"). So just saying E/c^2 is an inertial mass doesn't work. It makes even less sense as a source of gravity, especially since the source of gravity isn't some single mass number or energy density, but the entire stress energy tensor.

> I am not disagreeing with mainstream physics when I refer to the mass defect

Binding energy is a real thing, but that does not mean the invariant mass of an electron changed in the hydrogen atom. Invariant mass is not additive. Consider the system of two particles of invariant mass M1 and M2; the system does not necessarily have invariant mass M1+M2. It is completely unclear to any of us how your hydrogen atom example supports your claims, and you refuse to give math to clarify anything.

> I am not disagreeing with mainstream physics when I [...] point out that "invariant" mass varies.
> Please refer to mass in general relativity.

You don't even seem to understand what the issue that article is referring to. The issue is that in curved spacetime it is possible to not have a time-like killing vector, in which case we have locally conserved energy and momentum, but not a global one. Note that the Schwarzschild metric does have a time-like killing vector.

Do you at least understand that the invariant mass of a particle falling into a blackhole is indeed constant?

The action for a free particle, even in curved spacetime, is just:
S = integral m dtau
where tau is the length along the worldline. If the invariant mass changed as you claimed, then we'd have something like:
S = integral m(tau) dtau
with the mass changing along the path.

What you are saying is not mainstream physics.

Again, I ask, not rhetorically,
What would it take for you to consider that you are wrong?

LFB: relativistic mass is a measure of energy. The more energy you have the more gravity you get. Binding energy does mean the mass of the electron is reduced, just as the total mass of two colliding planets M1 and M2 is reduced when the kinetic energy is radiated away. There is no actual "negative energy", merely less positive energy with its mass equivalence. The hydrogen atom supports my claims because the mass of a body is a measure of its energy-content and If a body gives off the energy L in the form of radiation, its mass diminishes by L/c².

I understand what mass is general relativity is referring to, just as you understand non-invariance of its length, the invariant mass and conservation of energy. Just as you understand that when you drop a 1kg brick into a black hole the black hole mass increases by 1kg. That is mainstream physics. Re your question:

"What would it take for you to consider that you are wrong?"

A demonstration from you that when you lift a brick, you don't do work on it. Only you do, don't you? You add energy to that brick. And the mass of a body is a measure of its energy-content. Isn't it? Or is that not mainstream either?

> "Binding energy does mean the mass of the electron is reduced"

The invariant mass is not changed.
Let's look at it this way.

The energy levels of the hydrogen atom are proportional to m_e m_p / (m_e + m_p). Since the mass of the proton is so much more than the mass of the electron, this means the energy levels are very close to being proportional to the mass of the electron.

If the mass of the electron changed, this would be seen in the experimental evidence for hydrogen spectroscopy which is incredibly precise.
Also, note that when calculating the energy levels theoretically, a constant electron mass is assumed -- and this matches experiment.

You understanding of physics theory is wrong. And your ideas don't match experiment. You continuing to repeat your statements without further explanation is not helping, so we need to go to the math.

If you don't start actually showing math for your invariant mass claims, by answering Henry's repeated requests, then this will just start going in circles. So please show your math and respond to Henry. It is clear discussing on a layman's level will probably not get us anywhere. So let's look at the math of a particle moving in a Schwarschild background.

Doug,
It looks like the current idea won't work, but I have a suggestion that may help.

What if you make newtonian gravity like EM, so that changes propagate at a finite velocity. You could even just use Maxwell's equations. But change the force law so that like charges attract.

You'll get ma = m (gravity field stuff), so the m will still drop out. So it appears to satisfy the equivalence principle and so allow a geometry only description. And since it was inspired by EM, you get lorentz invariance and coordinate independence for free!

Thanks for sharing your ideas. Even when they don't work out, they make people think, and that's valuable. Your gravity posts and the amazing discussion from the last article got me to the idea I put there. So it is inspired by you. It seems like the next logical step. I don't know how to analyze it though.

fly ...:

Your suggestion sure would seem to make a lot of sense, and would certainly appear to be the simplest possible direction in which to find a formulation for Newton's Universal Law of Gravitation that may be consistent with the Lorentz transformations of (Special) Relativity.  I wonder why Einstein didn't think to try that?

Now, I'm not sure whether you are aware, nor do I know how many other readers are aware, but Einstein actually published a number of his "false starts" and "dead ends" along the way on his path toward a relativistically consistent formulation of Newton's Universal Law of Gravitation.

You see, curved space-time was the furthest thing from Einstein's mind when he first embarked upon his quest for a relativistically consistent formulation of Newton's Universal Law of Gravitation.  All he really wanted (at least at first) was simply a form of Newton's Universal Law of Gravitation that's consistent with his Theory of Relativity (what we now refer to as Special Relativity).*

OK.  Well, even if we suppose that Einstein missed such a simple potential solution as you have proposed, what about the issues that others have run across in trying to include gravity in Quantum Mechanics (QM)?

I know that many readers, here, are aware of at least some of the struggles between gravity (especially General Relativity) and QM.  Researchers have found that they can handle all "forces" that act in a manner similar to Maxwell's equations of Electromagnetism (EM).  In general, these are called Yang-Mills theories (and EM is the simplest of such).

So, wouldn't it make sense that if gravity could be formulated in a manner such as you have suggested, that this would solve this dilemma as well?  So why have these researchers not thought of such a simple potential solution, or tried such?

Now, lest you think I am making some form of "appeal to authority" sort of argument to dissuade you (and/or Doug) from pursuing what "must be" or "is obviously" a "fruitless" avenue of research, let me assure you that I believe that even pursuing research directions that result in "dead ends" is eminently worthwhile (at least so long as large societal resources are not involved in a pursuit that has already been well shown to be "barren"**).

Even if you, or someone, can find research that already shows why your simple potential solution cannot work, it can still be instructive for you or Doug to make the attempt; to learn from first-hand experience why such has not been pursued; to not just take someone's word for why it "doesn't work"—not even Einstein's word.

Sometimes, the journey is more important than the result.

David

*  However, admittedly, fairly early on, he did begin to wonder whether he might be able to find a more "general" form of "relativity" that would be as independent of ("absolute") acceleration as (Special) Relativity is of "absolute" velocity.

**  I make a definite distinction between the pursuit of research that ends up with a negative, or otherwise "disappointing" result (such as, say, the LHC discovering no new physics); vs. pursuing research directions that have already been well shown to be devoid of promise, for a given purpose.  Often, the negative, or otherwise "disappointing" results are just as valuable as finding what one hoped to find—in fact, such may be, arguably, even more valuable!