Gravity and Time Dilation: Length Contraction and Mass Inflation?

[HarryWertM]

According to SR theory, relative motion leads to
-length contraction
-mass inflation
-time dilation.

In GR theory, gravity leads to time dilation. Does it also lead to
-length contraction?
-mass inflation?

 

[ia_]

Special relativity is a subset of general relativity. In a region of space without large masses, the geometry of that space will become flat. That is, it will reduce to the minkowski metric of special relativity. So, all of the effects that occur in special relativity can also occur in general relativity.

In addition, there are a host of new, but it some cases analogous effects. Perhaps the most often observed new effect is that time runs more slowly for objects in a large gravitational potential. This is magnified in the case of a black hole. The rate of progression of time for an object falling into a black hole (as viewed by a distant observer) will slow to zero as that object approaches the event horizon.

For GPS satellites, which orbit the Earth at a high velocity, both special and general relativistic effects occur:

effect due to sr:
As GPS satellites move fast relative to us, their time is dilated, and runs more slowly, by about 7 microseconds/day.

effect due to gr:
Because gps satellites are further away from the center of the Earth's gravitational field, their time runs faster than ours, by about 45 microseconds/day.

According to wikipedia (http://en.wikipedia.org/wiki/Global_Positioning_System#Relativity), if neglected then these effects would cause uncertainties in position to grow by around 10km/day.

There are dozens of other new effects: frame dragging, event horizons, the penrose process, cauchy surfaces, schwarzschild and kerr black holes, expanding cosmologies, even messier differential equations, and so on:D

 

[Altabeh]

In GR theory, gravity leads to time dilation. Does it also lead to
-length contraction?
-mass inflation?

In GR there are no such things as "length contraction" and "mass inflation" caused by gravity but rather now there exists a new concept "curvature" by which one can feel the presence of a gravitational field and matter.

AB

 

[Frame Dragger]

Special relativity is a subset of general relativity.

Really? Funny how Einstein seemed to START with Special Relativity, then moved on to GR. Are you positive you're not just blowing smoke?

@Altabeh: Hmmmm... do you mind if I quote you to a friend of mine who's been struggling with this concept? You said that very succinctly!

 

[Altabeh]

@Altabeh: Hmmmm... do you mind if I quote you to a friend of mine who's been struggling with this concept? You said that very succinctly!

Yeah sure!

AB

 

[Janus]

Really? Funny how Einstein seemed to START with Special Relativity, then moved on to GR. Are you positive you're not just blowing smoke?

Yes, really. Einstein started with SR because it was easier to start with the limited "special" case. But he also knew that the theory would not be complete until it could be further expanded to deal with the more "general" case.

 

[George Jones]

Special relativity is a subset of general relativity.

Really? Funny how Einstein seemed to START with Special Relativity, then moved on to GR. Are you positive you're not just blowing smoke?

I.e, Einstein condsidered more general situations for which special relativity is a special case. If, in general relativity, one considers the special case of a simply connected spacetime with g_{\mu \nu} = \eta_{\mu \nu}, one gets Minkowski spacetime. In this case, Einstein's equation is 0 = T_{\mu \nu}, so energy and matter have to be (in some sense) negligible ("test" particles only).

 

[Frame Dragger]

I understand, but I don't think you can call a theory published while GR was still being formulated a "subset". I understand the physics, I disagree with the semantics. To be blunt, Einstein's first paper describing SR was published in 1905, and GR formulated between '07-'15.

I wouldn't call it a subset of anything, just "The Special and General Theories of Relativity".

 

[George Jones]

I understand, but I don't think you can call a theory published while GR was still being formulated a "subset".

Why not?

I understand the physics, I disagree with the semantics.

To me, the phrasing doesn't seem so bad. It is clear what is meant.

To be blunt, Einstein's first paper describing SR was published in 1905, and GR formulated between '07-'15.

For me, the dates are irrelevant.

I wouldn't call it a subset of anything, just "The Special and General Theories of Relativity".

Again, I don't think that the terminology is so bad.

 

[dx]

Electrostatics was formulated before the full electromagnetic theory of Maxwell. Yet, electrostatics is logically a special case of electrodynamics, in the sense that under specialized conditions, electrodynamics reduces to electrostatics. The relationship between special relativity and general relativity is the same.

 

[Frame Dragger]

I don't see how this argument over semantics, which I will gladly cede, can go anywhere useful. I aknowledge the point that General Relativity is a complete theory in which there is a "Special" case, as a subset. I would point out that a person in 1906, would of course have no way of appreciating that. It is logically organized in the manner you describe, but that isn't how it evolved. ia_'s language assumed prior knowledge that some people may not have on an educational site, not an entirely uncommon occurrence.

There are some brilliant people here, but the communication skills are occasionally lacking in terms of liasing between common parlance and various formalism and terms of art of mathematics and physics. This does lead to some extended misunderstandings and pointless detours such as this.

This is a situation in which, as with dx's example: there's more than one way of establishing a logical hierarchy. There is a historical context, in which theories evolve from one another, and then the formal (and useful) manner of classifing them ad hoc. I'm not disputing the latter, but let me ask you this: How would you establish a hierarchy for the major branches of mathematics?

@George Jones: "Again, I don't think that the terminology is so bad." Fair enough; I don't think the terminology is that great.

 

[HarryWertM]

So back to my original question - is there length contraction and/or mass inflation in GR - Altabeh said:

"In GR there are no such things as "length contraction" and "mass inflation" caused by gravity

Then there must be no such things as length and mass, or they do not contract and inflate.

Which?

 

[Altabeh]

I understand, but I don't think you can call a theory published while GR was still being formulated a "subset". I understand the physics, I disagree with the semantics. To be blunt, Einstein's first paper describing SR was published in 1905, and GR formulated between '07-'15.

I wouldn't call it a subset of anything, just "The Special and General Theories of Relativity".

Most physicists first get involved with a very limited and simple theory and then they try to expand the ideas brought up in the initial theory into a big framework wherein not only will the first theory work well, but the new theory will be based on some more general ideas that lead to the initial ideas in special cases. It is not at all a bad way to keep a theory going deep into broader areas of knowledge by first starting from a simple thing though this could sometimes lead to nowhere if the developed ideas are not as much compatible to the initial ideas as possible! For example, the idea of "curvature" in GR is compatibly reducible to the old implication of "force", as assumed by Newton. Sometimes there are no conditions by which the theory can be given a simplified, reduced or even generalized form. Google can help you find some examples of this.

AB

 

[HarryWertM]

Thanks anyway.

 

[Altabeh]

So back to my original question - is there length contraction and/or mass inflation in GR - Altabeh said:

Then there must be no such things as length and mass, or they do not contract and inflate.

Which?

I think my post can't be put in a clearer form: Length cannot contract and mass cannot inflate due to gravity in GR.

AB

 

[Frame Dragger]

So back to my original question - is there length contraction and/or mass inflation in GR - Altabeh said:



Then there must be no such things as length and mass, or they do not contract and inflate.

Which?

Are you by chance thinking of the old sci-fi standby, the "blueshifted 'front' of a near-c 'craft', versus the apparent long redshifted 'tail' of it? That is an effect perceived by the observer, and doesn't imply a real change in length or mass. In essence, you're seeing light "stack" in the front, and "extend" in the rear, but that's the LIGHT, not the "craft".

I'm not even sure what you mean by "mass inflation", unless you think that the mass of an object approaching 'c' becomes "infinite", which is a common misconception. Once again, you have to remember which effects are the result of a change in something, and which are merely observational artifacts.

@Altabeh: Yes, theories usually evolve from the simple to the more complex; and that new theory should incorporate the old, in which case from one point of view SR is clearly a subset of GR. That said, even though GR is the framework which explains SR and more, one did not evolve as a subset of the other, but rather GR is an expansion and extension (among others things) of GR.

As I said, this is a semantic issue, with two possible views on the subject. I'll admit that mine is less useful in this context, but also it won't mislead newcomers to the subject as to how the theories were developed. To me, a subset represents a group "B" containing elements derived or taken from group "A". For that to occur, group A needs to exist for a subset to emerge, rather than B leading to A and because A encompasses B, it's being called a subset.

As I said earlier, I would call them "SR and GR", not "GR and its subset", or "SR and its superset". There is no context in which identifying them as two theories, one leading to another, isn't preferable to "subset"

 

[cesiumfrog]

Are you by chance thinking of the old sci-fi standby, the "blueshifted 'front' of a near-c 'craft', versus the apparent long redshifted 'tail' of it? That is an effect perceived by the observer, and doesn't imply a real change in length or mass. In essence, you're seeing light "stack" in the front, and "extend" in the rear, but that's the LIGHT, not the "craft".

I'm not even sure what you mean by "mass inflation", unless you think that the mass of an object approaching 'c' becomes "infinite", which is a common misconception. Once again, you have to remember which effects are the result of a change in something, and which are merely observational artifacts.

A common misconception? Would you not concede that the inertia of an object (the quantity of external momentum impulse that it takes to alter its velocity a particular amount) increases without limit as the object approaches the speed of light? (After all, this is the mechanism that prevents us accelerating it to superluminal velocities. Let's just leave aside questions of precisely what the term "mass" should refer to.)

Sure, purely optical distortions also exist, such as Terrel rotation. But the reality of length contraction is the first lesson taught on SR (regarding a barn and ladder, unfortunately under the guise of a paradox involving details of material acceleration). Regardless of how your sci-fi craft seems visually, the fact is that physical synchronised measurements of its extent (say, made by attendants along a station as the craft passes against some of them) would confirm contraction (and note that this is predicted by GR as well as by SR, regardless of whatever point Altabeh is trying to make), just as surely as atomic clocks left in different floors of a building do confirm the time-dilation predicted in GR.

To put that another way, since the shape of atoms and molecules is basically dependent on Maxwell's theory (which exhibits Lorentz symmetry), whilst you may be familiar with notionally-spherical electron-orbital shapes for atoms at rest, the correct actual solution for the shape of the electron-cloud of a fast moving atom is more like a pancake.

 

[Frame Dragger]

A common misconception? Would you not concede that the inertia of an object (the quantity of external momentum impulse that it takes to alter its velocity a particular amount) increases without limit as the object approaches the speed of light? (After all, this is the mechanism that prevents us accelerating it to superluminal velocities. Let's just leave aside questions of precisely what the term "mass" should refer to.)

Sure, purely optical distortions also exist, such as Terrel rotation. But the reality of length contraction is the first lesson taught on SR (regarding a barn and ladder, unfortunately under the guise of a paradox involving details of material acceleration). Regardless of how your sci-fi craft seems visually, the fact is that physical synchronised measurements of its extent (say, made by attendants along a station as the craft passes against some of them) would confirm contraction (and note that this is predicted by GR as well as by SR, regardless of whatever point Altabeh is trying to make), just as surely as atomic clocks left in different floors of a building do confirm the time-dilation predicted in GR.

To put that another way, since the shape of atoms and molecules is basically dependent on Maxwell's theory (which exhibits Lorentz symmetry), whilst you may be familiar with notionally-spherical electron-orbital shapes for atoms at rest, the correct actual solution for the shape of the electron-cloud of a fast moving atom is more like a pancake.

Really?! Wow, talk about being 180 degrees off target. Well, in that light, I'm going to skip to that portion of MTW and learn what I thought I knew. Thanks for the correction, and lesson. By the way, if it isn't too much trouble what would be a toy solution for a simple fast-moving atom (H, He)?

 

[yuiop]

According to SR theory, relative motion leads to
-length contraction
-mass inflation
-time dilation.

In GR theory, gravity leads to time dilation. Does it also lead to
-length contraction?
-mass inflation?

In SR an observer measures:

1) The length of rod parallel to the relative relative motion v, to be length contracted by sqrt(1-v^2).

2) The length of a rod transverse to the relative motion v, to be the same length as when it is at rest.

3) The rate of a clock with relative motion v, to be time dilated by 1/sqrt(1-v^2).

In GR an observer at infinity in Schwarzschild coordinates measures:

A) The length of stationary vertical rod at radial coordinate r, to be length contracted by sqrt(1-2GM/r).

B) The length of a stationary horizontal rod at radial coordinate r, to be the same length as a local rod.

C) The rate of a stationary clock at radial coordinate r, to be time dilated by 1/sqrt(1-2GM/r).

As you can see from the above, 1,2 and 3 are closely related to A, B and C respectively. Is that the sort of relationship you are looking for?

 

[HarryWertM]

"Kev": Yessir! Outstanding answer!

 

[Altabeh]

Sure, purely optical distortions also exist, such as Terrel rotation. But the reality of length contraction is the first lesson taught on SR (regarding a barn and ladder, unfortunately under the guise of a paradox involving details of material acceleration). Regardless of how your sci-fi craft seems visually, the fact is that physical synchronised measurements of its extent (say, made by attendants along a station as the craft passes against some of them) would confirm contraction (and note that this is predicted by GR as well as by SR, regardless of whatever point Altabeh is trying to make),

I just saw you mentioned in passing that "length contraction" is also predicted by GR. I assume you know that this is not the case which was asked about by the OP. If you entered GR so as to be able to for example make the length of a body contract in a given gravitational field due to the effects of gravity, I'd be more comfortable with the word "impossible" if assigned to the purpose of your entry. In GR length contraction is predicted if one comes in the inertial frames in Minkowski spacetime by reducing the new theory to SR. Using the principle of equivalence this could be done in the context of GR in a very small region but this again calls for gravity to disappear. Some eccentric sort of gravitational length contraction exists that only reveals itself when a 'ruler' is set at fixed (t,\theta,\phi). Such a contraction with this assuption is not realistic but just theoretical because the time can't be made constant in reality for a rigid body falling inward the black hole radially.

AB

 

[yuiop]

"Kev": Yessir! Outstanding answer!

Thanks! I have another "relationship" you might be interested in. Consider a particle dropped from infinty towards a (non-rotating, uncharged) massive body. Let us say the freefall velocity of the particle at radial coordinate r is measured to be v by a stationary observer a r. The time dilation that the particle would have in SR terms due to its freefall velocity is 1/sqrt(1-v^2/c^2) and this is equal to the time dilation experienced by a stationary particle at r due to gravitational time dilation, i.e. 1/sqrt(1-2GM/(rc^2)). Just in case you are wondering, the freefalling particle experiences both velocity and gravitational time dilation equal to 1/sqrt(1-v^2/c^2)*1/sqrt(1-2GM/(rc^2)).

To make this clearer, here is a numerical example considering a clock freefalling from infinity. Let us say the stationary observer at r measures the freefall velocity of the falling clock to be 0.8c. The free falling clock would then be running 0.6 time slower than a stationary clock at r, due to kinematic time dilation. The stationary clock at r would be running 0.6 times slower than a clock at infinity, due to gravitational time dilation. The free falling clock at r would then be running 0.36 times slower the clock at infinity, due to both effects.

 

[HarryWertM]

Foof. Now I know nothing. Just found this arxiv article [peer reviewed?]

http://arxiv.org/abs/0910.2298"

which states clearly that lengths EXPAND under acceleration [which is equivalent to gravity]. The expansion is given by L[1+[at/c]^2]^1/2. [Eq. 4 pg. 2] Maybe I have observer / accelerated reversed?

 

[Altabeh]

Foof. Now I know nothing. Just found this arxiv article [peer reviewed?]

http://arxiv.org/abs/0910.2298"

which states clearly that lengths EXPAND under acceleration [which is equivalent to gravity]. The expansion is given by L[1+[at/c]^2]^1/2. [Eq. 4 pg. 2] Maybe I have observer / accelerated reversed?

No, the authors state that this idea only works for accelerated frames in SRT and so there's no sign of gravity to think about it in this not peer-reviewed and badly typed article!

AB

 

[yuiop]

Foof. Now I know nothing. Just found this arxiv article [peer reviewed?]

http://arxiv.org/abs/0910.2298"

which states clearly that lengths EXPAND under acceleration [which is equivalent to gravity]. The expansion is given by L[1+[at/c]^2]^1/2. [Eq. 4 pg. 2] Maybe I have observer / accelerated reversed?

Hi Harry,

If you are new to relativity, then you managed to find just about the most confusing article a beginner could wish to find. Let me try and clear up some of the confusion. Let us start with some things you might be confortable with. I hope you agree that when a rod (in its relaxed state) is moving relative to you and parallel to its length, that you would measure its length to be shorter than when it is at rest with respect to you (in its relaxed state). I have stressed the "relaxed state" of the rod because when an inertially moving rod is measured to be length contracted in relativity, it is assumed to be in an unstressed state (i.e. neither physically stretched nor physically compressed) and the length of the rod is assumed to remain constant from the point of view of an observer moving with the rod. Now if we were to nail down one end of the rod and accelerate the other end of the rod towards the fixed end, the rod would be compressed, but this would not be be relativistic length contraction and the change in length is due to stress forces which could be measured. If the free end is accelerated away from the fixed end the rod would be stretched and become longer but we would not call that relativistic length expansion. Now when a free rod is accelerated in a manner that does not put any stresses on the rod (no stretching or compressing forces) then it length appears to remain constant according to an observer co-moving (co-accelerating) with the rod and appears to be getting shorter according to an observer that remains in the original rest frame of the rod before it started accelerating. This form of acceleration, that does not put any physical stresses on the rod is called Born rigid acceleration. Since the rod appears to getting shorter in the original rest frame of the rod, the back end of the rod appears to be catching up with the front end of the rod, so Born rigid acceleration requires the back end of the rod to be accelerating slightly faster than the front end of the rod. Any other kind of acceleration puts physical stresses on the rod that physically stretch or compress the rod and this is a separate issue from pure relativistic length contraction. As I mentioned before if I nail down one end of the rod and the accelerate the other end away from the fixed end, the rod gets stretched and I can claim that as an example of a rod getting longer when accelerated. A less extreme example is to accelerate the back end at the same rate as the front end. This is not Born rigid acceleration and put physical stresses on the accelerating rod and physically stretches the rod. In this case an observer moving with the rod sees the rod getting longer (because it is being stretched) and the observer that remained in the orginal rest frame of the rod sees the length of the accelerating rod remain constant. I think this is basically what is happening in the paper you have cited, but I have not read it in much detail. You have effectively stumbled across a version of Bell's rocket paradox which is not the easiest of paradoxes to understand, so I hope my explanation is not too confusing.

 

[HarryWertM]

No, the authors state that this idea only works for accelerated frames in SRT and so there's no sign of gravity to think about it in this not peer-reviewed and badly typed article!

SRT plus acceleration equals GRT. Equivalence principle.

I hope my explanation is not too confusing.

Indeed, after reading Wikipedia your clarification is as good as I could find. I think I did have unaccelerated observer frame confused with accelerated frames.

 

[starthaus]

A) The length of stationary vertical rod at radial coordinate r, to be length contracted by sqrt(1-2GM/r).

Apologies for being pedantic, not "vertical at radial coordiante r" but "alongside the r direction"

B) The length of a stationary horizontal rod at radial coordinate r, to be the same length as a local rod.

Same correction as above, "perpendicular on the r direction" (i.e. lying in the (theta,phi) plane)

 

[Altabeh]

SRT plus acceleration equals GRT. Equivalence principle.



Indeed, after reading Wikipedia your clarification is as good as I could find. I think I did have unaccelerated observer frame confused with accelerated frames.

Excuse me? Why on Earth did you reach the reasoning that if "acceleration" is added to SRT, then the resulting theory must be GRT? Even in SRT we have sometimes acceleration involved with the setup of frames and this doesn't mean the content of theory changes to GRT! The equivalence principle says that both the gravitational and inertial accelerations are equivalent in a sufficiently small region of a curved spacetime. What does this have to do with that article? There authors state clearly that the theory they are bringing this up in is SRT not GRT!

AB

 

[starthaus]

Just in case you are wondering, the freefalling particle experiences both velocity and gravitational time dilation equal to 1/sqrt(1-v^2/c^2)*1/sqrt(1-2GM/(rc^2)).

Dear kev,

How can that be? If you neglect the rotational terms, you obviously use the Schwarzschild metric with d(theta)=d(phi)=0.

Then, you get:

d\tau/dt=sqrt((1-r_s/r)-(v/c)^2/(1-r_s/r))=sqrt((1-r_s/r-v/c)(1-r_s/r+v/c)/(1-r_s/r)) even if we know that:1-r_s/r=1-1-2GM/(rc^2)This is very different from your result. How did you make the jump from the above formula to your formula?

 

[Passionflower]

A common misconception? Would you not concede that the inertia of an object (the quantity of external momentum impulse that it takes to alter its velocity a particular amount) increases without limit as the object approaches the speed of light? (After all, this is the mechanism that prevents us accelerating it to superluminal velocities. Let's just leave aside questions of precisely what the term "mass" should refer to.)

You are completely wrong. Answers like this cause people to be utterly confused about relativity.

Any beginner in learning about relativity should readily understand that just because something is moving near light speed with respect to an accelerating object does not mean that it interferes with the ability for that object to accelerate.

The relativistic mass of an object does not undergo a proper acceleration, the rest mass does. Regardless of the rate or duration of proper acceleration any object will measure a speed of light of c forever.