Space and time as fields of mass-energy

[The Count]

In my post graduate course, several years now, our professor in field theory have mentioned that in field theory the fields of mass-energy seem to be space and time themselves, like electric and magnetic fields in ElectroMagnetism. Specificaly he said that "the problem is that in ElectroMagnetism the fields are like the actors on the scene, but in gravity the fields seemed to be the scene itself! So its difficult to handle them...".

This means that as an electric charge creates electric field (and if moved magnetic field), at the same way mass can "create" space and time. I am aware of the bending of space through mass, but not the "creation" of it.

I am having trouble finding literature that explains that and maths to proove that.

Can anybody help?

 

[haushofer]

I can imagine; you won't find it because it's not true.

In Gen.Rel. spacetime consists of a manifold ("=" all the possible events) plus a metric g. This metric is a field, analogous to the electromagnetic 4-vector A. Just like the electromagnetic field equations tell you that the dynamics of A is constrained by a conserved current J, the Einstein equations tell you that the dynamics of g is constrained by a conserved energy-momentum tensor T. This T depends on all the other fields in spacetime.

I wouldn't even know what it means to "create" spacetime, except maybe through the big bang in the case of a finite universe. But that's a process not described by GR.

 

[Orodruin]

Electric charge does not create the electric field. The electric field is there regardless of whether the charge is or not. What the charge does is to change the way the field behaves.

 

[The Count]

Thanks for answering so fast.

https://www.physicsforums.com/members/haushofer.20128/

In Gen.Rel. spacetime consists of a manifold ("=" all the possible events) plus a metric g. This metric is a field, analogous to the electromagnetic 4-vector A. Just like the electromagnetic field equations tell you that the dynamics of A is constrained by a conserved current J, the Einstein equations tell you that the dynamics of g is constrained by a conserved energy-momentum tensor T. This T depends on all the other fields in spacetime.

We agree up to the 4-vector A.
The 4-vector in EM shows electric and magnetic field how to form, in the same way that the metric shows spacetime how to form-bend?
Or we have to put in the notion of 4-vector for gravity the Ricci tensor too?
And why can't we assume, that as the 4-vector in EM "generates" electric and magnetic field, in the same analogy, the 4-vector for gravity "generates" space and time?

I am sorry for this going to Gen.Rel. topic, I thought it had to do with Quantum Field Theory or similar.

I wouldn't even know what it means to "create" spacetime, except maybe through the big bang in the case of a finite universe. But that's a process not described by GR.

Electric charge does not create the electric field. The electric field is there regardless of whether the charge is or not. What the charge does is to change the way the field behaves.

Can't we assume (even hypothetically) that we have "empty space" (isolated, or EM shielded) or vacuum and we introduce a charge? This could possibly mean that we "generate" electric field (and magnetic if we move it).
In the same way if we have vacuum (in a sense of not empty space, but of "no space"(?), if such thing has any sense) and we "introduce" a mass do we "create" space?

 

[haushofer]

Thanks for answering so fast.

We agree up to the 4-vector A.
The 4-vector in EM shows electric and magnetic field how to form, in the same way that the metric shows spacetime how to form-bend?

Yes. But note: spacetime consists of a manifold PLUS the metric. Not just the metric! This subtlety is also crucial in understanding the 'hole'- argument, btw.

Can't we assume (even hypothetically) that we have "empty space" (isolated, or EM shielded) or vacuum and we introduce a charge? This could possibly mean that we "generate" electric field (and magnetic if we move it).
In the same way if we have vacuum (in a sense of not empty space, but of "no space"(?), if such thing has any sense) and we "introduce" a mass do we "create" space?

No. If you are far away from an energy/mass distribution (and the cosm.const. is zero), spacetime becomes flat. This means that spacetime is given by a manifold PLUS the metric of spacetime, which is the Minkowski metric.

Mass and energy are always contained IN spacetime.

 

[vanhees71]

Thanks for answering so fast.

Can't we assume (even hypothetically) that we have "empty space" (isolated, or EM shielded) or vacuum and we introduce a charge? This could possibly mean that we "generate" electric field (and magnetic if we move it).
In the same way if we have vacuum (in a sense of not empty space, but of "no space"(?), if such thing has any sense) and we "introduce" a mass do we "create" space? [/SIZE]

No, we can't, at least not if the current theory about the electromagnetic field is correct, and we have no reason to doubt that. In it's classical version it has been discovered by Maxwell in 1865 (building on the collected knowledge about electromagnetism by 1-2 generations of physicists before him, particularly Faraday, who discovered the very important (if not the most important) notion of fields in the course of his comprehensive experimental studies of electromagnetic phenomena) and has been a great success story since then. It's modern version is part of the Standard Model of elementary particle physics, i.e., a relativistic quantum field theory, but it's better to understand the classical part better first. The quantum field theory is more subtle to interpet right, and without a clear understanding of the classical field theory you have no chance to ever understand it.

So let's discuss within the classical theory (which by the way is indeed relativistic although discovered by Maxwell quite some decades before relativity has been discovered by his successors like Poincare, Lorentz, FitzGerald, and of course Einstein). First there's some empirical input to any theory in physics. If there's no such empirical input carefully worked into the theory, you can as well forget it. It might be a nice mathematical idea but usually it's no physics. Electrodynamics, as all successful theories in physics, is built on observed facts about nature. The most fundamental observation is that there exists a feature of material bodies that we describe as electric charge, which comes as positive and negative charge (nowadays we know that it comes in an integer number of discrete charge units, the elementary charge, which is the charge of the electron (negative) or proton (positive, exactly opposite to the electron charge!). This is, however, neglected in the classical theory, and you deal with the charge ##q## of macroscopic bodies as a continuous quantity. For bodies at rest these charges manifest themselves in terms of electrostatic forces, ruled by Coulomb's Law, i.e., the force is in magnitude proportional to the product of the charges of the bodies and inversely proportional to the square of the bodie's distance.

The modern picture for this is, however, that the force acts on the charge ##q_1##, because the presence of charge ##q_2## necessarily means that there is an electromagnetic field associated with it. If ##q_2## is at rest with respect to an observer for a very long time, there's only an electrostatic Coulomb field, and ##q_2## feels a force because of the presence of this field at the place where ##q_2## is located (i.e., it's a local description of the force, not like the old-fashioned action-at-a-distance picture in Newtonian mechanics, which would lead to tremendous trouble in the context of relativity).

The next observation is that if the charge is moving (in the most simple case with constant velocity) in addition to the electric field you also observe a magnetic field, which leads to forces on other moving charges. The total force on a charge ##q## is given by the Lorentz force,
$$\vec{F}=q \vec{E}+\frac{q}{c} \vec{v} \times \vec{B},$$
where ##\vec{E}## and ##\vec{B}## are the electric and magnetic field (or better said, the electric and magnetic components of the electromagnetic field as observed in some frame of reference), ##\vec{v}## is the velocity of the charge, and the fields are to be evaluated at the momentaneous position of the charge at the present time (again with respect to a fixed reference frame and realizing the idea of local interactions due to the notion of fields).

Now there are also mathematical laws for the electromagnetic field itself, the Maxwell equations, which describe, how the electromagnetic field is related to the charge and current density of the matter, which are the sources of these fields (+some constraint equations to close the set of equations of motion for the fields for given charge-current distributions of the matter). As it turns out, these equations are only mathematically sensible (or consistent) if and only if electric charge is strictly conserved, i.e., you cannot create a new net amount of charges but only separate charges from each other. So you cannot just have vacuum and all of a sudden put somehow a charge in it. The only thing you can do is to bring some charged body at a certain place and look at the field around it, and this field in principle depends on the entire history of the charge-current distribution! So there's always a field around the charges, and it's governed by how these charges move.

On the other hand the Lorentz force tells us how, on the other hand, the electromagnetic field created by some charged matter acts as force on other charged matter, accelerating it. Now, Maxwell's equations also predict that when charges are accelerated, i.e., when the charge-current distribution becomes time dependent, the electromagnetic field is not just the "Coulomb field" of the moving matter but that electromagnetic waves are radiated (this is closely connected to the locality of the theory and relativity), which means that the electromagnetic field itself carries energy, momentum, and angular momentum (all of which are conserved only for the entire system of charge-current distributions and the electromagnetic field as a whole). This implies that there is energy and momentum carried away from the matter (i.e., it's motion) in terms of the electromagnetic field, and this means this motion is damped. Also electromagnetic radiation can be absorbed by matter (e.g., leading to heating it up like in the microwave oven were electromagnetic radiaton energy is absorbed by the food you like to cook).

This leads to a quite complicated self-consistent problem of the dynamics of electromagnetic fields and charge-current distributions, which gave the greatest minds of physics (including Lorentz, Abraham, Dirac, Schwinger, Feynman, and many more) a big headache. At least for the motion of "point-like particles" the solution of this problem, including the radiation reaction on the accelerated motion of charged particles, is not yet been found, and one might be in doubt, whether such a solution exist at all. So far we have only approximations, which work quite well, at least well enough that we can construct highly precise particle accelerators like the LHC at CERN.

Also electromagnetism has not only brought the discovery of relativity but also of quantum theory, because the absorption and emission processes of electromagnetic fields with matter cannot be fully understood by the classical theory. One famous problem related with it is the "black-body spectrum", i.e., the electromagnetic spectrum emitted by a hot cavity with walls kept at a certain temperature for a long time so that the entire system of walls and the electromagnetic radiation within the cavity comes to thermal equilibrium. It has been clear already around the mid of the 19th century that this is a universal spectrum, independent of the material the cavity is made of. The accurate observation of this spectrum by physicists at the Physikalische Reichsanstalt in Berlin around 1900 (with the aim to set a standard for the then upcoming electric light bulbs, i.e., a very practical purpose!) lead finally Planck to the discovery of quantum theory, i.e., that the exchange of energy and momentum between the electromagnetic field and matter has to be described by a radically different theory than classical physics. The so far final theory on this has been finally formulated already in 1925 by Heisenberg, Born, Dirac, Schrödinger, and many others, but that's another story.

 

[The Count]

Starting I want to thank you vanhees71 for all your briefing in EM .

So you cannot just have vacuum and all of a sudden put somehow a charge in it.

You are right that if we put a charge means that we transferred it with its field at the same time. Although, one could use production of matter-antimatter couples. I believe that even though we have two opposite charges (ex. e⁺ and e^- ) and out of the system the total charge remains 0, locally in space when we separate them, we have charge created out of "nowhere" (as far as electric charge and field are concerned, because I am aware that we need energy to create the coupling), so should the field should be created at once.
Except if we assume that the energy that created the coupling appearence is photons, which means EM waves, which means the EM "field" again is already there.
Can't we have different kind of energy like mass turning to energy, or other particle combination that gives just the annihilation and not use the photons?

the Lorentz force tells us how, on the other hand, the electromagnetic field created by some charged matter acts as force on other charged matter, accelerating it.

You also use the term "created".

No. If you are far away from an energy/mass distribution (and the cosm.const. is zero), spacetime becomes flat. This means that spacetime is given by a manifold PLUS the metric of spacetime, which is the Minkowski metric.
Mass and energy are always contained IN spacetime.

I already mentioned if we had to take into account the Ricci tensor (which is the manifold if I understand correctly) as seen in the equation below.

But even if he have to take into account the Ricci tensor too (which is also derivatives of spacetime), it doesn't change the question which was not answered btw, if I am not mistaken:

And why can't we assume, that as the 4-vector in EM "generates" electric and magnetic field, in the same analogy, the 4-vector for gravity "generates" space and time?

Yes, I have to say that the particle interaction theory somehow makes it more complicated, and in a way cancels the terms "field" (at least in the classical sense). It seems more easy to imagine Einstein's sheet of spacetime to be the "field" of a mass, than a graviton to interact with a massive particle and change it's space and time, whatever that means.

also I am not sure that this question was answered finally

in field theory the fields of mass-energy seem to be space and time themselves, like electric and magnetic fields in ElectroMagnetism

 

[The Count]

I hope I was not rude to anybody.
I would appreciate any help.

 

[vanhees71]

I don't see, where you may have been rude :-).

Anyway, it's not so clear what you mean by your statement that "in field theory the fields of mass-energy seem to be space and time themselves, like electric and magnetic fields in ElectroMagnetism".

You wrote down the Einstein-field equations for the gravitational field and claimed you can read off something in this direction, but that's not what the equations tell you. They rather tell you that the energy-momentum tensor of matter and radiation (the right-hand side of the equation) is the source of the gravitational field in general relativity. This is so, because the left-hand side of the equation is nothing else than a partial differential equation of 2nd order for the pseudometric ##g_{\mu \nu}##, which is for the gravitational field in some sense analogous to the vector potential of the electromagnetic field. They coupling between fields and matter/radiation by the way is necessarily universal, i.e., there's only one graviational constant ##\kappa## for all kinds of energy-momentum tensor, no matter of what makes this energy and momentum.

The similarities between electromagnetism and gravity as described in GR is no accident but due to the fact that both theories are gauge theories with the difference that electromagnetism is an Abelian gauge symmetry (with the gauge group U(1)) and GR a non-abelian one (with the gauge group GL(4)). For details about how to interpret GR as gauge theory, see

Ramond, Pierre: Field Theory: A Modern Primer, 2nd edition, Addison-Wesley, 1989