Is/are there any invariant OBJECTS in relativistic? Is there a substratum?

[jwdink]

Hi, I'm currently writing a paper on Relativity, which mostly uses original papers of Einstein. For this reason, I have little idea what the ultimate fallout of all his upheaval is. I am aware that electromagnetic fields become "shadows" of the complex mathematical entity called the "Electromagnetic tensor," length and time become "shadows" of invariant "spacetime" intervals, gravitational mass becomes inertial mass following warped geodesic/inertia lines (expressed, again, by tensors), and inertial mass becomes another measure/property of energy. Since energy is not invariant, this implies that mass is not invariant, which implies (in my opinion) that "matter" is not invariant. It seems that--while the laws are becoming more and more invariant--the invariant, persisting subject which underlies all of this begins to disappear. Our only measure of matter has become a measure of energy content, and you can therefore not call "inertial mass" the same thing as "stuff," since you can increase the inertial mass of something by putting more energy into it-- without necessarily putting more "stuff" into it, as in the case of a tensed string having more mass than a loose string. Meanwhile, subatomic particles can pop in and out of existence.

SO! I'm wondering if, like the invariant space-time interval, there's something analogous in relativistic physics for a persisting "object" that we can say does NOT pop in and out of existence depending on reference frames etc. Obviously, it would have to be at least four-dimensional. Is there anything like this, or do we really lose any applicability we had of our notion of "stuff/substance/substratum" in the metaphysical sense?

Thanks!

 

[tiny-tim]

Hi jwdink!

… Since energy is not invariant, this implies that mass is not invariant, which implies (in my opinion) that "matter" is not invariant.

Rest-mass is invariant.

And rest-mass is what "matter" is.

And although yes, energy is not invariant, the energy-momentum 4-vector is invariant (or, more precisely, covariant), so the energy-and-momentum of matter is covariant, and in that sense also "matter" is invariant.

I'm wondering if, like the invariant space-time interval, there's something analogous in relativistic physics for a persisting "object" that we can say does NOT pop in and out of existence depending on reference frames etc.

Sorry, nothing "pops in and out of existence" depending on the reference frame.

 

[jwdink]

Rest-mass is invariant.

And rest-mass is what "matter" is.

If you tense a string, doesn't it have more mass than when it is relaxed? This does not sound like the concept of "matter" that is never created or destroyed.

Sorry, nothing "pops in and out of existence" depending on the reference frame.

A given electric or magnetic field does, no? I'm stationary relative to a magnet, and experience no electric force. I'm moving relative to a magnet, and do experience an electric force. I ascribe energy to this field. Therefore, the energy is measured different depending on the reference frame. I think we both agree that 3-dimensional things can pop in or out of existence depending on reference frame, while 4 dimensional things don't.

so the energy-and-momentum of matter is covariant, and in that sense also "matter" is invariant.

...say more...

 

[tiny-tim]

A given electric or magnetic field does, no? I'm stationary relative to a magnet, and experience no electric force. I'm moving relative to a magnet, and do experience an electric force. I ascribe energy to this field. Therefore, the energy is measured different depending on the reference frame.

If you look at a disc side-on, it looks one-dimensional, but if you move a little, it looks two-dimensional.

That doesn't mean the extra dimension has suddenly "popped into existence".

The electric field of the magnet gradually changes. In one frame of reference, it happens to be zero. That doesn't mean it doesn't exist in that frame, any more than the speed of something doesn't exist in a frame in which it's stationary.

If you tense a string, doesn't it have more mass than when it is relaxed? This does not sound like the concept of "matter" that is never created or destroyed.

You're deliberately confusing mass and "matter".

If you tense a string, it has more energy than when it is relaxed, and that energy came from you.

Yes, matter can be converted into energy (and vice versa), but that has nothing to do with frames of reference, and therefore in no way contradicts the invariance or covariance of matter.

 

[Altabeh]

...say more...

In the theory of GR, the covariance ¹ of energy can be checked by the well-established law of conservation of energy $$T^{{\mu \nu }}_{; \nu }=0$$ where $$T^{\mu \nu }$$ being the second-rank matter tensor and ';' is the covariant differential operator in any curved spacetime. In SR, ';' is replaced by ',' that is the partial differential operator in Minkowski spacetime.

To learn more about the conservation laws of GR, take a look at the chapter 8 of Papapetrou's "lectures on GR".

¹(This term is mostly preferred by authors active in GR, in place of 'invariance')

 

[jwdink]

You're deliberately confusing mass and "matter".

Not sure I am. You said rest mass is identical with matter. I'm giving an example of an object's rest mass being augment without adding more "matter" to it. Therefore, the concept of rest mass cannot refill the gap vacated by our previously naive view of "matter" meanings "stuff." We'll need some four-dimensional mathematical entity, such as, perhaps...

$$T^{{\mu \nu }}_{; \nu }=0$$ where $$T^{\mu \nu }$$ being the second-rank matter tensor

Which is CRAZY!

If you look at a disc side-on, it looks one-dimensional, but if you move a little, it looks two-dimensional.

That doesn't mean the extra dimension has suddenly "popped into existence".

Unless you believed that the world was necessarily one-dimensional. But again, I think we're in agreement here. I'm careful in the paper for the confusion about things popping in and out of existence to imply they are "shadows" of something else. But, if we forced ourselves to look at them as three dimensional, then they do indeed have a rather curious type of existence.

 

[Dale]

Since energy is not invariant, this implies that mass is not invariant, which implies (in my opinion) that "matter" is not invariant.

There are two definitions of the word "mass" in relativity. One is called "relativistic mass" it is what you are referring to above, it is simply the total energy scaled to units of mass. The other is called "invariant mass" or "rest mass", this is the invariant Minkowski norm of the four momentum so all reference frames agree on it. When modern physicists, especially particle physicists, use the unqualified term "mass" they usually refer to this second definition.

I think it is a mistake to equate matter with mass, but if you do, simply use the usual invariant mass and then you have invariant matter.

 

[Altabeh]

Which is CRAZY!

Which part of it does sound crazy to you and Why!?

 

[jwdink]

Which part of it does sound crazy to you and Why!?

Hahaha, crazy in a good way. I should have said "shocking/incredible/surprising/impressive" etc.

But also crazy in a crazy way. The whole point of my paper is using Einstein and Kant to ask: "didn't we have an a priori concept of some substratum, instantiated in nature as some three-dimensional matter than never disappears? then how did Einstein use empiricism to contest this notion, and force us to make any sort of substratum four-dimensional?" I think we did have a notion of three-dimensional matter, and therefore it is pretty surprising that Einstein was able to replace it with four-dimensional tensors.

There are two definitions of the word "mass" in relativity. One is called "relativistic mass" it is what you are referring to above, it is simply the total energy scaled to units of mass. The other is called "invariant mass" or "rest mass", this is the invariant Minkowski norm of the four momentum so all reference frames agree on it. When modern physicists, especially particle physicists, use the unqualified term "mass" they usually refer to this second definition.

But even invariant mass can change without having more "matter" put into it, as the example of a tensed string demonstrates. So clearly rest mass is not synonymous with a naive notion of "matter" meaning "quantity of stuff." There's no substratum in three dimensions, just energy.

 

[Dale]

But even invariant mass can change without having more "matter" put into it, as the example of a tensed string demonstrates. So clearly rest mass is not synonymous with a naive notion of "matter" meaning "quantity of stuff."

I agree. That is why I said that I think it is a mistake to equate matter and mass.

 

[DrGreg]

There seems to be some confusion here between "invariant" and "conserved". I once said in another thread...

It is perhaps worth mentioning two concepts, "conservation" and "invariance", which can sometimes be confused with each other.

A conserved quantity is something measured by a single observer that doesn't change over time; for example it has the same value before and after a collision, and typically it is the sum of several measurements, e.g. of multiple particles. Examples are energy (a 1D number), momentum (a 3D vector), four-momentum (a 4D vector), all when there are no external forces, of course. In Newtonian physics, mass is also conserved. In relativity, relativistic mass may be conserved (in the absence of any other form of energy) but rest mass isn't.

An invariant quantity is a single measurement whose value all observers agree upon, i.e. a frame-independent value. Examples are proper time, (scalar) proper acceleration, and rest mass. Or anything that can be expressed in the form $g_{ab}U^aV^b$ (where U and V are genuine 4-vectors).

So, energy and momentum are both conserved but neither is invariant. In relativity, rest mass is invariant but not necessarily conserved across multi-particle interactions. (In Newtonian physics, mass is both conserved and invariant.)

 

[jwdink]

So, energy and momentum are both conserved but neither is invariant. In relativity, rest mass is invariant but not necessarily conserved across multi-particle interactions. (In Newtonian physics, mass is both conserved and invariant.)

Is anything both? That was my original question. Is there some tensor to express an invariant conservation law?

 

[nicksauce]

You can use Killing vectors to construct conserved quantities. If [itex]k^{u}[/tex] is a Killing vector, then $k_{\mu}p^{\mu}$ is conserved along geodesics. (And it is invariant since it is a scalar). This can be viewed as the general relativistic version of Noether's theorem.

 

[jwdink]

I will google those things.

 

[nicksauce]

Essentially (and roughly speaking), killing vectors are symmetries. Some simple examples: If a system is invariant under time translations, (1,0,0,0) is a Killing vector. If a system is axisymmetric then (0,0,0,1) is a Killing vector in Spherical co-ordinates t,r,theta,phi.

I don't know if you know about Noether's theorem, but it is an extremely important theorem in Classical Mechanics that says that symmetries lead to conserved quantities.

 

[jwdink]

You can use Killing vectors to construct conserved quantities.

Wait, sorry, I missed this. Conserved quantities? That could plausibly be different from what I'm asking.

Let's look at it this way. Back in the good old days, it all made sense: everything was material, and this material "stuff" has certain properties, which we reified into various "laws." The stuff was never created or destroyed.

Now, we've learned that a concept of three-dimensional "stuff" or "material" is untenable. Einstein is looking for the invariant LAWS of the universe-- but does that mean the "stuff," the subject of these laws, has vanished? Is the universe just a bunch of disembodied laws? Or is there still a plausible subject, which, though only representable in our minds as a complex four dimensional entity (perhaps a tensor), is nevertheless the thing which our science is about, and fills up our world. It is what the objective world consists of. OR: have we just turned the objective world into a bunch of equations with different quantities, and some of these quantities are intuitable to us, and give the illusion of being "stuff"?

 

[Altabeh]

Wait, sorry, I missed this. Conserved quantities? That could plausibly be different from what I'm asking.

Let's look at it this way. Back in the good old days, it all made sense: everything was material, and this material "stuff" has certain properties, which we reified into various "laws." The stuff was never created or destroyed.

Now, we've learned that a concept of three-dimensional "stuff" or "material" is untenable. Einstein is looking for the invariant LAWS of the universe-- but does that mean the "stuff," the subject of these laws, has vanished? Is the universe just a bunch of disembodied laws? Or is there still a plausible subject, which, though only representable in our minds as a complex four dimensional entity (perhaps a tensor), is nevertheless the thing which our science is about, and fills up our world. It is what the objective world consists of. OR: have we just turned the objective world into a bunch of equations with different quantities, and some of these quantities are intuitable to us, and give the illusion of being "stuff"?

I think now you have started to talk about the philosophy of "matter", as you call it, in some true but not logical sense, "stuff", in the universe rather than finding out what laws are behind all those physical concepts that we are used to assign to matter to meet the demands of nature in terms of mathematics, which is reasonably translated 'physics'. I don't think Einstein has changed our 'matter' or 'view' and 'beliefs' toward it nor has he anything else. Let's say, "Hey man, you need to change the angle you look at physics through." I got to say that if Einstein had not felt the 'stuff' with his tactile perception, he would not have ended up bringing us to a new and more realistic word than Newton's word. You better suspect the idea of a 'world made of strings' that is in my eyes the most hated one in the whole physics. Is it plausible and meaningful to you having something like a string as the principal 'stuff' of the universe!?

 

[Dale]

Back in the good old days, it all made sense: everything was material, and this material "stuff" has certain properties, which we reified into various "laws." The stuff was never created or destroyed.

Now, we've learned that a concept of three-dimensional "stuff" or "material" is untenable.

I don't see how any of this has really changed through Einstein and Minkowski. All they did was to show that things that we had previously separated, like time and space or energy and momentum we now found to be unified into bigger concepts like spacetime or four-momentum. This unification, imo, didn't destroy any of the previous concepts of matter, it only enhanced them by showing a previously unseen connection.

Einstein really did not change our perception of "stuff", that was Schrodinger, Dirac, et al.

Btw, for any conserved 4-vector the norm of that 4-vector is both conserved and invariant, e.g. the rest mass of any system is both conserved and invariant.

 

[Altabeh]

Einstein really did not change our perception of "stuff", that was Schrodinger, Dirac, et al.

Even the latter ones didn't change our perception of "stuff". We just had not discovered what that "stuff" was in essence up until the time Democritus, Dalton, Thompson, Rutherford, Bohr, et al laid out the nature of "tiny stuff" and argued that there is something strange going on as if we had no awareness of until again Schrodinger, Dirac and ... started to give away some rules that govern the tiny stuff and interactions between its elements. But I don't see how this has something in common with conservation, invariance and covariance. I think our sort of natural perception is going to spoil if string theory finds its way to experiments. There I would say "yeah, see Witten and et al turned the world upside down..."!

 

[Dale]

Even the latter ones didn't change our perception of "stuff". We just had not discovered what that "stuff" was

I am not going to get involved in a purely semantic argument like this. Say it however you prefer.

 

[ZirkMan]

Now, we've learned that a concept of three-dimensional "stuff" or "material" is untenable. Einstein is looking for the invariant LAWS of the universe-- but does that mean the "stuff," the subject of these laws, has vanished?

Einstein just showed us that this "material stuff" is only Energy and it's conserved but not invariant. A 2D shadow does not cease existing when you realize there must be a 3D object that casts it.

Is the universe just a bunch of disembodied laws? Or is there still a plausible subject, which, though only representable in our minds as a complex four dimensional entity (perhaps a tensor), is nevertheless the thing which our science is about, and fills up our world. It is what the objective world consists of. OR: have we just turned the objective world into a bunch of equations with different quantities, and some of these quantities are intuitable to us, and give the illusion of being "stuff"?

We do not know a definite answer to this. The hope of physics is to find the former through the latter, I guess ;). Though, perhaps more than 4 dimensional entities will be found to exist to explain the observed reality.

 

[jwdink]

Einstein just showed us that this "material stuff" is only Energy and it's conserved but not invariant. A 2D shadow does not cease existing when you realize there must be a 3D object that casts it.

I guess I'm objecting to something that is not invariant being called "stuff," in the sense of "substratum."

 

[agnostikos]

Why does cosmic background radiation not provide a 'preferred reference frame'?

 

[ZirkMan]

I guess I'm objecting to something that is not invariant being called "stuff," in the sense of "substratum."

But you cannot change a frame without a change in total Energy. Therefore the non-invariance of Energy (as the substratum) is a direct consequence of the very definition of invariance.

 

[atyy]

But even in the "old way of thinking" with vectors there were no "invariants", eg. (Ex=1,Ey=2,Ez=3), but we can just rotate the axes so that (Ex=2, Ey=-1, Ez=3) is a different set of numbers representing the same "thing". So the electromagnetic tensor is no different, instead of 3 numbers, its 16 numbers, that's all.

 

[Altabeh]

Einstein just showed us that this "material stuff" is only Energy and it's conserved but not invariant. A 2D shadow does not cease existing when you realize there must be a 3D object that casts it.

Einstein actually showed us the "stuff" is energy and Weyl showed that how a 2D space cannot contain "stuff'" if being discussed in the sense of mass. Energy in GR is always conserved but it is not invariant. The covariance of Energy tensor does not account for its being invariant in the sense that it remains the "same" in another frame. This can be immediately obtained using Einstein's field equations. You can see on the left-hand side R_ab is non-invariant under the change of coordinates system so is energy tensor on the other side.

AB