What is the mass of a string near c ?

[Mike2]

What is the mass of a string near "c"?

If the mass of a string/particle increases with the vibrational frequency of the string, and if time slows down when you travel near the speed of light, then wouldn't it appear to a stationary observer that the frequency of the string and thus the mass of the string was decreasing as it neared the speed of light? Yet this would seem to contradict the predictions of Special Relativity which states that mass would increase to infinity as one approaches the speed of light. Which one is it?

 

[da_willem]

the predictions of Special Relativity which states that mass would increase to infinity as one approaches the speed of light

mass will remain constant, whatever the speed of an object, even in special relativity. It is true that it becomes increasingly harder to accellerate an object near the speed of light. So you could say it's inertial mass increases.

If the mass of a string/particle increases with the vibrational frequency of the string

If this one of the 'laws' of string theory it probably is about invariant mass which does not change with speed. So independent of the speed of the string it will have a certain mass and frequency.

 

[Mike2]

If the mass of a string/particle increases with the vibrational frequency of the string

If this one of the 'laws' of string theory it probably is about invariant mass which does not change with speed. So independent of the speed of the string it will have a certain mass and frequency.

Of course the rest mass would be invariant in a frame of reference moving with the same velocity. The whole question is how would a stationary observer perceive the mass of a string/particle. Normally, for point particles, the mass appears to increase as it approaches the speed of light as viewed by the stationary observer, right? And also clocks appear to run slower for a speeding objects as viewed from a stationary observer, right? How then does the frequency of a string moving near c appear to a stationary observer? And would this affect the stationary observers perception of the string's mass? Thank you.

 

[da_willem]

Normally, for point particles, the mass appears to increase as it approaches the speed of light as viewed by the stationary observer, right?

No, as I tried to point out to you this is not true. This is an old interpretation of the relativistic equations and deals with a concept called 'relativistic mass'. The modern interpretation is a different one and there is no place for relativistic mass anymore. So even if you move with a certain speed relative to a particle it's mass is the same. And the string relation you mention is probably about this invariant mass. So there's no problem here.

 

[Mike2]

No, as I tried to point out to you this is not true. This is an old interpretation of the relativistic equations and deals with a concept called 'relativistic mass'. The modern interpretation is a different one and there is no place for relativistic mass anymore. So even if you move with a certain speed relative to a particle it's mass is the same. And the string relation you mention is probably about this invariant mass. So there's no problem here.

You haven't mentioned what the old idea of relativistic mass is replaced with.

And has time dilation also been "reinterpreted". Or is the mass of a string-particle no longer a function of vibrational frequency? As you can see, relativistic mass is not relevant to the question, only time dilation and string-mass due to frequency are relevant.

 

[aekanshchumber]

look, frequency of the string does not decide the mass or something like that for any thing, it only desides the charecter of the element it is vibrating for. one particular frequency decide only on particular partile.

 

[Mike2]

look, frequency of the string does not decide the mass or something like that for any thing, it only desides the charecter of the element it is vibrating for. one particular frequency deside only on particular partile.

So would it then appear to a stationary observer that a moving string was changing continuously from one TYPE of particle into another?

Even so, then what effect would space contraction have on the characteristics of a string-particle?

 

[aekanshchumber]

Mike2
First of all, no one can see a string vibrating. every thing is visible (to human eye) because of photons and photon themself aremade up of string. if someone however is vibrating with same frequency and is in same phase he might observe that no string is vibrating so nothing apears to him ( as t-rex can't see any thing still, can say). But to a stationary observer both the string and the observer might seems like same partilces

 

[da_willem]

You haven't mentioned what the old idea of relativistic mass is replaced with.

And has time dilation also been "reinterpreted".

Time-dilation still stands. But in special relativity the definition of e.g. momentum had to be revised to preserve the law of conservation of momentum. In special relativity this law still holds if you take for momentum $\gamma mv$ instead of mv. At first this was interperted as an increase in mass ($m= \gamma m_0$)so the old formula, $m_0 v$ would still hold but now it's just a revised definition of momentum.

And I'm not sure (as I haven't studied string theory ) about the frequency of a string and if it is about a real vibration or about some abstact property called vibration...so if time dilation has any effect on the frequency.

 

[sol2]

"String theory is an idea for describing elementary particles in a way that differs a little bit from what people have done in the past. Previously, it seemed that elementary particles could be viewed as being points, with no size at all, just mathematical points. And this idea ran into trouble when we tried to include gravity in the theory."

http://www.hyper-mind.com/hypermind/universe/content/gsst04/anim1.gif

"So, in string theory, what we have done is generalize the idea of point particles to particles that have extensions, namely they are lines, they have no thickness at all, but are one-dimensional curves. In the most promising theories they are loops, to be precise. So, string theory is a theory in which the elementary particles are loops of one-dimensional elementary objects."

http://www.hyper-mind.com/hypermind/universe/cast.htm

 

[Mike2]

http://www.hyper-mind.com/hypermind/universe/cast.htm

It would seem to me at this time, that if particles really are strings, or loops, or membranes of higher dimensions that "vibrate" in some way with a rest frequency, then whatever characteristic that is dependent on that vibration frequency would change with speeds near the speed of light. This would be a way to confirm that particles are indeed the result of vibrating extended objects. So the questions are: what characteristic of these extended objects are dependent on frequency, and can we measure a change in that characteristic as the object approaches the speed of light. If no changes at all are observed between moving and still particles/strings/membranes, then string theory will have been falsified.

So what particle characteristics do change with speed near c? There is the mass, what about spin? Does that change near c? If infinite mass were the result of no vibration, then this would be consistent with relativity. But if mass appears to decrease as frequency decreases, then this would appear to contradict relativity that says appearent mass increases near c. Then again, maybe there is a sudden decrease in mass just before you reach c, i don't know. And maybe there is a decrease in mass in dense gravity wells. That would at least eliminate an infinitely dense singularity at the big bang and perhaps black holes as well. Thanks.

 

[sol2]

http://physicsweb.org/objects/world/13/11/9/pw1311091.gif

a) Compactifying a 3-D universe with two space dimensions and one time dimension. This is a simplification of the 5-D space*time considered by Theodor Kaluza and Oskar Klein. (b) The Lorentz symmetry of the large dimension is broken by the compactification and all that remains is 2-D space plus the U(1) symmetry represented by the arrow. (c) On large scales we see only a 2-D universe (one space plus one time dimension) with the "internal" U(1) symmetry of electromagnetism.

It does not say there is no energy. In fact, this is what we look at.

The huge Planck mass means that if such a string theory provides the correct description of quantum gravity, then everything we see today is essentially massless as far as the theory is concerned.

The whole notion of extra dimensions has its origin in the search for a unified theory of the forces observed in nature. The story began in the 1860s with the unification of the electric and magnetic forces by James Clerk Maxwell. As well as the extraordinary prediction that light is an electromagnetic wave, Maxwell's theory had a hidden property that was not realized until much later. It has what we now call a "gauge symmetry".

Gauge symmetry can be visualized in the following geometrical way. Suppose that every charged particle has associated with it an arrow that can rotate round in a circle like one of the hands of a clock. This rotation does not take place in the 3-D space that we observe, so the circle is - for the moment - purely mathematical, and the symmetry, known as U(1), is deemed "internal". The symmetry principle states that the absolute positions of these arrows can never be determined. Moreover, the symmetry is said to be "gauged" or "local" - meaning that the definition of absolute arrow position can change with time and location. Allowing such variations introduces a spurious current unless we add an extra ingredient to exactly compensate for it. This extra mathematical ingredient is the electromagnetic field.

The presence of this field explains the physical properties we associate with electromagnetism. For example, the field carries pulses of energy that we observe as particles of light - photons - and the exchange of photons results in the net electromagnetic force between charged particles.

In the 1920s Maxwell's unification of electricity and magnetism, together with Einstein's new general theory of relativity, inspired Theodor Kaluza and Oskar Klein to suggest that it might be possible to unify electromagnetism and gravity in an overarching geometrical scheme involving extra dimensions.

General relativity is a wonderful example of a geometrical theory. It too is derived from a local symmetry, known as Lorentz symmetry, that involves the four dimensions (three space plus one time) of everyday experience. In this case, velocities are like the arrows of the U(1) symmetry. So Lorentz symmetry incorporates the fact that the results from physical experiments are independent of the direction from which we view them and of our velocity. General relativity makes the symmetry local and, as for electromagnetism, that requires a field - which in this case is the geometry of space-time itself. Local "ripples" in space

http://physicsweb.org/article/world/13/11/9#pw1311091

Hyperspace, by Michio Kaku Pg 9

Since the theory was considered to be a wild speculation, it was never taught in graduate school; so young physicists are left to discover it quite by accident in their casual readings. This alternative theory gave the simplest explanation of light; that it was really a vibration of the fifth dimension, or what used to be called the fourth dimension by the mystics. If light could travel through a vacuum, it was because the vacuum itself was vibrating, because the “vacuum” really existed in four dimensions of space and one of time. By adding the fifth dimension, the force of gravity and light could be unified in a startlingly simple way...

What is the true vacuum?

http://www.damtp.cam.ac.uk/user/gr/public/images/inf_old_inf.gif

First order phase transitions proceed by bubble nucleation. A bubble of the new phase (the true vacuum) forms and then expands until the old phase (the false vacuum) disappears. A useful analogue is boiling water in which bubbles of steam form and expand as they rise to the surface.

Go backwards to the beginning of the universe, what was our universe's shape

You have to undertand where the "Planck epoch" is. The singularity would take you here and it is very spread out. This information exists around us now.

So, in string theory, what we have done is generalize the idea of point particles to particles that have extensions, namely they are lines, they have no thickness at all, but are one-dimensional curves. In the most promising theories they are loops, to be precise. So, string theory is a theory in which the elementary particles are loops of one-dimensional elementary objects.

 

[Schneibster]

You know, I came across this thread in a google for the answer to the same question this IP asks, and I see no answer here. I see lots of nice descriptions of string theory, but frankly, I have seen lots of others and these don't help answer this question at all, at all. The closest anyone came was a "probably," and some kind of description of "relativistic mass" that appears to ignore the equivalence principle, that inertial mass and gravitational mass are equivalent. This is not much help. Neither is the imputation that the poster of the IP is stupid.

So what gives? What changes in the relativistic string's equation under the Lorentz transform? Is there a change in the equation of vibration when the transform is applied? And if there is, what is the physical meaning of that change?

 

[Schneibster]

Bump, am I to conclude that no one knows? This should not be that hard a question. Is there any answer?

 

[nightcleaner]

Ok I'll give it a shot. Objects in string theory are multidimensional, in the sense that their behavior cannot be sufficiently described in three dimensions of space and one of time. This is not a proven fact, it is part of the theory.

String theorists believe their extra dimensions are spatial. So they should also be affected by relitive motion. However this is inconvenient, because relativity only deals with three space and one time dimension. Therefore it is impossible to know how the "other" space dimensions behave under acceleration, where we might expect to see Lorenz contraction. So your conclusion that no one knows is probably correct. If you think the question is not hard, you should answer it yourself.

My personal theory, which does not yet have falsifiable expression, is that the other dimensions in string theory could be thought of as time dimensions. Maybe this line of thought will turn out to be useful, maybe not. Fortunately my income does not depend on the outcome, which leaves me free to use my idle time to speculate.

I hope you will uncurl your lip long enough to contribute something other than mere mockery to the discussion.

Be well,

nc

 

[selfAdjoint]

Nightcleaner, multidimension relativity is perfectly possible. The squared metric would be $$c^2x_0^2 - x_1^2 - ... -x_9^2$$, for 10-dimensional superstrings. The Lorentz transformations would be 10 X 10 matrices, and work just the same as the 4 X 4 ones we are more familiar with. The extra dimensions are spatial in string theory and relativity is built into the theory

 

[nightcleaner]

Thank you, selfAdjoint.

Perhaps you can help me with a similar question...do all dimensions contract equally under acceleration, or are there differences in amount of contraction depending on the orientation of the dimension to the direction of acceleration? In other words, if the object is traveling in the x direction, does it contract equally in x as in y and z?

Perhaps my confusion is due to the idea that the contraction is a rotation. If there is a rotation, is there an axis of rotation? And if there is an axis of rotation, would the dimension of the axis not change differently from the dimensions bearing a tangential angle to the acceleration?

BTW, I picked up A. Zee's nutshell book, and am very pleased to find that I can fool myself into believing I understand something of what is presented in English, altho of course I am still struggling with the maths. But, progress is progress. Thank you again for all your help.

Richard

 

[selfAdjoint]

Thank you, selfAdjoint.

Perhaps you can help me with a similar question...do all dimensions contract equally under acceleration, or are there differences in amount of contraction depending on the orientation of the dimension to the direction of acceleration? In other words, if the object is traveling in the x direction, does it contract equally in x as in y and z?

All dimensions work just the same. In fact you can have a relative velocity an any direction, say $$\pi x + e y - \sqrt{2} z$$ and the contraction would happen along that line with no contraction in any direction perpendicular to that line. And with more dimensions you just have more variables.

Perhaps my confusion is due to the idea that the contraction is a rotation. If there is a rotation, is there an axis of rotation? And if there is an axis of rotation, would the dimension of the axis not change differently from the dimensions bearing a tangential angle to the acceleration?

The Lorentz transformation (remember time and space both come into it) is a "rotation" in Minkowski space. The minus sign between the time and space coordinates of Minkowski space means that the sines and cosines of a 3-D rotation matrix turn into hyperbolic functions: cosh and sinh. These aren't anything new, Euler defined them back in the 18th century, but they are unfamiliar to most people because most introductory math courses don't cover them. They have a family resemblance to sin and cos through identities, but their performance is different; unlike sin and cos they can go to infinity. There are textbooks describing all this. A classic is Taylor and Wheeler's Spacetime Physics, which I recommend.

BTW, I picked up A. Zee's nutshell book, and am very pleased to find that I can fool myself into believing I understand something of what is presented in English, altho of course I am still struggling with the maths. But, progress is progress. Thank you again for all your help.

Richard

I am glad you are feeling you're making progress! Sometimes studying by yourself can be the worst drag in the world, but if you push on you come out the other side and it's all exciting again. Expect to go around with Zee a couple of times, as you learn other things something you slid over suddenly seems meaningful - or maybe problematical! This is the glory of learning.

 

[nightcleaner]

All dimensions work just the same. In fact you can have a relative velocity an any direction, say $$\pi x + e y - \sqrt{2} z$$ and the contraction would happen along that line with no contraction in any direction perpendicular to that line. And with more dimensions you just have more variables.

Yes, I thought I remembered it that way, but doubted my memory for a moment. Just to confirm, for acceleration in x, the hyperbolic tangent affects the basis in x, not in y or z, and intermediate angles are affected only in their x component.

Then the question becomes, are there any other perpendicular basis lines? That is, are all the "extra dimensions" just mixes of x, y, z and t? Since t also experiences contraction, i presume that t is thought to be parallel to the line of travel. If the extras are just mixes of the usual 4 suspects, why do we need to call them extras? I have been trying to think of extra dimensions as extra basis lines, that is lines some of which at least would be perpendicular to the line of travel. This led me to wonder if some of the extra dimensions might be unaffected by the contraction, since they might not have an x component.

Then there is spin. If the spin axis is in x, a sphere becomes flattened at the poles. But what happens when the sphere is rotated in y while being accelerated in x? The x component of an object on the equatorial surface will vary between one and zero, having the value one twice in each rotation and the value zero twice in each rotation. But perhaps this expansion and contraction is not energetic, since it is occurring in spacetime? No local change over t would occur? Tidal and thermal forces would not appear?

Thanks for the encouragement.

Be well,

nc

 

[selfAdjoint]

Yes, I thought I remembered it that way, but doubted my memory for a moment. Just to confirm, for acceleration in x, the hyperbolic tangent affects the basis in x, not in y or z, and intermediate angles are affected only in their x component.

Then the question becomes, are there any other perpendicular basis lines? That is, are all the "extra dimensions" just mixes of x, y, z and t? Since t also experiences contraction, i presume that t is thought to be parallel to the line of travel. If the extras are just mixes of the usual 4 suspects, why do we need to call them extras? I have been trying to think of extra dimensions as extra basis lines, that is lines some of which at least would be perpendicular to the line of travel. This led me to wonder if some of the extra dimensions might be unaffected by the contraction, since they might not have an x component.

The other dimensions are in addition to the usual three space dimensions, so insead of saying x, y, and z, we say $$x_1, x_2, x_3, x_4, ...$$, as many as we need. They all combine to make multidimensional directions just as the x, y, and z do, and the direction is still a line in multidimensional spacetime so the Lorentz effect works along that direction just as it does on any direction in 3-space.

Then there is spin. If the spin axis is in x, a sphere becomes flattened at the poles. But what happens when the sphere is rotated in y while being accelerated in x? The x component of an object on the equatorial surface will vary between one and zero, having the value one twice in each rotation and the value zero twice in each rotation. But perhaps this expansion and contraction is not energetic, since it is occurring in spacetime? No local change over t would occur? Tidal and thermal forces would not appear?

The relativistic account of spinning bodies requires the full power of the Lorentz transforms; I don't think you can really work it out with separate length contraction and time dilation as in the case of a simple linear speed difference. But I'll try.

If the sphere in your example is spinning in y, then its equator lies in the x-z plane, and at rest would be a circle. Leaving acceleration (another complication) aside and just speaking of velocity in the x direction relative to some observer, that circle would appear flattened^*, and if the sphere were far enough from the observer that we could neglect parallax, so would all the "latitude" circles in planes parallel to the equatorial plane. Meanwhile the spinning would as you say produce oblateness so that in any plane through the y-axis the circular outline of the sphere becomes an ellipse; this widens out the equator circle, which (maybe) would scale up but still retain its flattened shape. So this non qualittative work through (which you should not rely on) suggest that the spphere will become something like a general ellipsoid, elliptical in the x-z, the x-y and the y-z planes.


^*Take the equation of the uncontracted circle to be $$x^2 + z^2 = 1$$, the unit circle at the origin in the x-z plane. If the velocity in the x-direction is +v then the x-contracted value relative to the observer is
$$\sqrt{1 - \frac{v^2}{c^2}}x$$

which we take to be uniform across the z diameter (this is the simplification that let's us use the simple formulas). So the contraction depends only on the speed, not on x, and we can write the equation of the contracted circle as
$$\gamma x^2 + x^2 = 1$$, where $$\gamma = \sqrt{1 - \frac{v^2}{c^2}}$$
and this is the equation of an ellipse.

 

[Schneibster]

Nightcleaner: thank you. Very much closer to a real answer than the original answer; I still have a question or two, but haven't had time to write them yet. By no means did your answer make me disdainful.
SelfAdjoint: thanks for your clarifications.