Can inertial reference frame have different sizes at least in theory?

[smm]

hi
can inertial reference frame be in different sizes at least in theory? i mean that can there be two different frames of reference in same place at same time, where the observer in both frames observes physical laws to be normal in their own frame of reference, but the other frame appears to be smaller than the other and both agree with this? is there some fact that deny this?

special relativity could give at least one restriction for such inertial obs. reference frames:
-if the frame is L times larger, its time must run L times slower- otherwise it would emit light at superluminal speed. right?

there is at least two open questions what comes into mind:

1 would change in size of reference frame cause some internal forces (inertia or tidal) in the inertial reference frame or not?

2 would larger reference frame have more energy than smaller?

yes -i know that in astronomy there aren't any known phenomena where the size of inertial reference frame would change into another or be another.

 

[Mentz114]

I think you might have a wrong notion of what a frame of reference is. It is not a physical object that can change in size. Have a look at the Wiki page

http://en.wikipedia.org/wiki/Frame_of_reference

 

[Dale]

can inertial reference frame be in different sizes at least in theory?

In theory the size of an inertial reference frame is infinite in all three dimensions of space and also infinite in time.

 

[smm]

hi!
thanks for the link:)

in special relativity, laws of physics are same in all frames of reference that have constant velocity , that is called a rest frame. and all rest frames are equal, that is the principle of relativity.

what i am asking by can there be reference frames in different size, is not the overall size, but that can there be reference frames that laws of nature follow different sizes or scale, or different unit lengths- while they may be both in same place and time.

for example can there be a frame of reference, whose unit length is say 1.2 meter instead of 1.0 meter - and everything in that frame of reference is 1.2 times larger - the size of atoms for example.

(another question related to this is that would it be possible that single frame of reference has different unit length at different distance from its origin, for example the unit length is say 0.7 or 1.5 in one distant point, while it is 1.0 in origin)

for example let's think of proton, that travels alone in space a long distance, and for some reason, its frame of reference expands to 1.2 times larger during its travel. That would mean that the protons size would expand 1.2 times larger, and its volume would be (1.2) ³ times larger

the special relativity theories principle of light velocity invariance would require that
if such frame of reference exists, its time must run 1.2 times slower. otherwise the proton would emit light at superluminal speed.

as i already wrote there are at least two questions :if this is possible in theory, would the energy of larger frame of reference be larger? also would the expansion of reference frame cause any inernal forces to reference frame.

i am not sure can some fact deny this, and i haventt ever heard anyone talking about this, but i know of course that this kind of phenomenom is unknown in physics or in astronomy. intuively i can think that the closest thing that could involve expansion of reference frame is ultrarelativistic cosmic rays that travel long distances and Hubbles law (in this case it would be expansion of light's reference frame) but i don't know enough about astronomy to think this trough - there i heard is 10 cosmological tests that can test different models of space.

 

[Mentz114]

I think you are describing curved spacetime. This is a very complicated field ( it is for me anyway ). The theory of curved spacetime is described in this Wiki article

http://en.wikipedia.org/wiki/General_relativity

 

[bcrowell]

In theory the size of an inertial reference frame is infinite in all three dimensions of space and also infinite in time.

In GR, I would say exactly the opposite. In GR, frames of reference are local, not global. They're infinitesimally small.

It seems like the OP's ideas are not clearly formulated. What s/he means could be equivalent to (a) the idea of a varying metric, (b) an unobservable change of coordinates, or (c) Weyl's original gauge theory, in which a clock can end up running at a different rate after being transported around a closed loop.

For info on Weyl's idea, see:

Weyl, "Space-time-matter," http://www.archive.org/details/spacetimematter00weyluoft p. 282

Eddington, "Space, time and gravitation: an outline of the general relativity theory," http://www.archive.org/details/mathematicaltheo00eddiuoft

 

[smm]

hi i try to put together the idea and question here again :
lets define that natural unit length of a reference frame at rest or constant speed is a unit length that all or at least most laws of nature follows.
the question is, can there coexist in same place two reference frames that have different natural unit length? can for example atom or partilce exist in reference frame that is at rest but it has say 1.2 times larger diameter than "normal" particle?
also, can natural unit length vary from place to place inside one reference fram?

 

[HallsofIvy]

No. If a person in reference frame "A" will see "B"s unit length as smaller than his and "B" will see "A"s unit length as smaller than his.

 

[smm]

hi! no, i don't mean lorenz contraction of relativistic moving reference frame!

i mean is it possible and if not, what facts denys that - at least in theory to exist of two reference frames in same rest speed side by side that has different natural unit length in all 3 dimensions -that both observers inside each frame measure all (or at least most) laws of natures laws to be normal in their own frame of reference but they observe natures laws follow different natural unit in other reference frames - and that they both agree what frame (and also what observer) is big and what is small.

at least special relativity's lights velocity invariance principle would require that if for example reference frames unit length is L times longer, its time must run L times slower - because otherwise atom in that frame would emit light at superluminal speed.

thanks for the link about that wileys idea. by the way i was thinking that if for example reference frame always expands gradually along any geodecic path in space, that would lead that object in circular orbit would gradually grow in size and its time would slow down - and i don't know what would happen to circular orbits in for example in galaxies if all constantly mobing reference frames has variations in natural unit length in astronomical scales... would the circular orbit radius change slowly?

 

[Dale]

In GR, I would say exactly the opposite. In GR, frames of reference are local, not global. They're infinitesimally small.

In GR I would agree. I took the question as a SR question. Perhaps the OP can clarify.

 

[bcrowell]

hi i try to put together the idea and question here again :
lets define that natural unit length of a reference frame at rest or constant speed is a unit length that all or at least most laws of nature follows.
the question is, can there coexist in same place two reference frames that have different natural unit length? can for example atom or partilce exist in reference frame that is at rest but it has say 1.2 times larger diameter than "normal" particle?
also, can natural unit length vary from place to place inside one reference fram?

This would happen in Weyl's gauge theory, described in #6. It doesn't happen in standard relativity.

 

[smm]

well i was thinking empty space iow special relativity to simplify the question

well i don't understand the math of general relativity theory. is the "size" of reference frames nature laws independent feature from gravity?
what would happen to gravity of some particles reference framef expands?
what would happen to object in circular orbit if its reference frame expands?
if every rest frame has slight variation in natural unit length in astronomical scale, would the circular orbits change slightly over time?

the energy of larger reference frame is not clear to me how it would be (if this larger and expansinng reference frame is even possible)- could expanding reference frame if its more energetic it could gain energy for free iow without doing any work- this could be possible if reference frame moves long distances in space where local natural unit length gets smaller and moving reference frames natural units does not - nothing happens to moving reference frame "inside it", observer inside it feels to be at rest, but it expands relative to local reference frame that has smaller natural unit length far away from starting point??

 

[strangerep]

snm,

One can augment the transformations of SR with so called "dilations" which alter the scale. But you keep saying "natural unit of length". The term "natural" in this context usually means "free from arbitrary choices", hence it would have to be an "invariant" length scale. (Cf. the speed of light $c$ is an "invariant" velocity scale, as far as we know.)

But if the length scale is invariant, then by definition it's the same for all observers, and then your question becomes self-contradictory. (I hope you can see what I mean.)

 

[smm]

sorry about my english, it may be that i may use some wrong words and wrong name- i don't know if it has better name, like "natural differences between natural laws" or "fixed set of physical constants" but i mean "natural unit" as unit, that most nature laws follows, like speed of light, magnitude of interactions, size of atom, size of bohrs radius, le broglie wavelength. These have certain fixed value in rest reference frame, independent from the observers choose of the units, right?
-whether the unit is 1 meter, 1 inch, 1 ångstrom, or 1 lightyear.

(i am asking can there be rest reference frame that have different length scales in theory, that is- its meter is say 1.2 times greater than normal meter - and atoms in this frame have for example bohr's radius 1.2 times larger etc -and can it exist together with normal reference frame in same place- clearly this is not known phenomenom in physics or in astronomy. i can think of at first sight two phenomenom what could have something to do with expansion of reference frame: hubbles law and ultrarelativistic cosmic rays. well, the third may be subatomic radiation but boy i don't really know particle physics well to say can for example gamma radiation have expansion in reference frame after it leaves from atom nucleus.

the light speed might be invariant in frames with different unit scale, then i think the time must run L times slower in L times larger reference frame)

thanks for the link about weyl gauge theory, ill read about this

 

[Dale]

I think what you are asking is whether or not a transformation of the following form represents a legitimate transformation between inertial reference frames:
$$\left( \begin{array}{c} T \\ X \\ Y \\ Z \end{array} \right) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & k & 0 & 0 \\ 0 & 0 & k & 0 \\ 0 & 0 & 0 & k \end{array} \right) \left( \begin{array}{c} t \\ x \\ y \\ z \end{array} \right)$$
Is that what you are asking?

 

[smm]

that is close, i am not very familiar of minkovskys matrix use, but i think the matrixs diagonal line would be

K,K,K,K that is, locally time in reference frame that is K times bigger runs K times slower
when the comparison is made between frames that have different size and are together
in one place in same rest frame. if the other frame is moving, then it has also relativistic
changes.

but there is one peculiar thing if you think of situation where natural unit length
varies from place to place and have different values in one rest frame.

The length of time unit can't vary inside one reference frame, otherwise the communication between different places would break causality - if the information is carried at speed c.
i have thought this a bit and it seems its illogical.

(i deleted one offtopic text)

-the time can't run at different rate inside one reference frame in different places, i think so but two different sized reference frames time can run at different rate when they are measured and compared in one point locally. am i right or is there logical failure in this?

when thinking this situation where natural unit length varies from place to place inside one reference frame the diagonal line would be 1,K,K,K or just K,K,K without the time element - but the local comparison between different sized reference frames matrix's diagonal line would be still K,K,K,K or is it 1/K K K K

if a traveler moves from origin to other place and his reference frame expands say K times in size relative to this one frame, and local observer in this distant place observes the traveler, the observer see the traveler being K times larger in size and his time would run K times slower. -and the traveler would see that local observer and everything in his frame locally is K times smaller, and his time would seem to run K times faster. velocity of light is in both frames same c and in fact all constant velocities are invariant if i assume that the time moves at different rate in different sized reference frame.

 

[mfb]

You can just re-label all physical units and corresponding constants. If you do it consistently, you get the same physics again, but Earth has a diameter of 1 meter, light needs 20ns to go around, or whatever. All observers will agree on the physical predictions, and all but you will agree that you use weird units. As you have to change the gravitational constant as well, the Planck units will change their numerical value. You will get the same value for the diameter of earth, measured in Planck units - there is no way to change that (with current physics).

 

[Dale]

that is close, i am not very familiar of minkovskys matrix use, but i think the matrixs diagonal line would be

K,K,K,K that is, locally time in reference frame that is K times bigger runs K times slower
when the comparison is made between frames that have different size and are together
in one place in same rest frame.

No problem, I can write it without using matrices as follows:
$T=kt$
$X=kx$
$Y=ky$
$Z=kz$

This is not a valid transformation between inertial frames. Consider Newton's 2nd law for a force, f, in the x direction:
$$m\frac{d^2 X}{dT^2} = m\frac{d^2(kx)}{d(kt)^2} = m\frac{k}{k^2} \frac{d^2 x}{dt^2} = \frac{1}{k} f$$

 

[smm]

yes the kinematics of larger reference frame would be different if its observed from
smaller reference frame.

i am not sure at all how the dynamics would change. i have guess that larger reference frame has K² times more energy and 1/K times slower energy transfer speed, and 1/K times smaller energy density. this comes from energy of harmonic oscillator that has energy K²kx²/2 if the string constant is same.

how does different reference frames interact? is the larger reference frame "stiffer" since its time run slower or is it weaker and more fragile? i assumed above that slow time makes reference frame "stiffer".

about kinematics:

-if the K times larger reference frame is different in its natural unit length K times greater and time rate that is K times slower and there are no other changes, then

-all time dependent functions would have transformation f(t)' = f(t/K) and all path dependent functions would have transformation f(l)'=f(Kl)

-for example harmonic oscillator would oscillate K times slower and it would seem to
dampen K times slower in larger reference frame.

-all functions time derivatives if measured from normal reference frame, would be 1/K times slower

the constant velocity would be invariant or same measured in both reference frames
constant acceleration would be 1/K times slower in larger reference frame.

i may do math mistake here, but would all higher order time derivatives of any time dependent function be then always 1/K times slower.

i don't know how the higher order derivatives would seem to change, do they have some other rule than this. this could be if there are some other changes in larger reference frame than these two.

about the dynamics:

the different sized reference frame could have some other changes if these two changes arent enough to make the dynamics laws to work right.

i have a rough guess that K times larger reference frame has always K² times more energy than normal reference frame in all energy forms (including mass?). for example nonrelativistic kinetic energy would be K²mv²/2
and the energy transfer rate could be K times slower, the energy density E/V K times smaller... this comes from dimensional analysis of energy of harmonic oscillator kx²/2
- if the string constant k is same then energy of harmonic oscillator is K² times larger. i don't have to take slowness of time into account since energy transfer is also 1/K times slower, right?

odd thing about energy conservation is that if the energy is K² times larger and the reference frame does not exerience any forces during expansion, then this energy is free - or if small reference frame expands to larger one, it would have K² times more energy without actually doing any work.
-but what are the limitations how the process can happen, is another question... it may be property of space that for example if reference frame moves long distances it may change larger proportional to the distance relative to local reference frame. this can happen if every rest frame has variation in natural unit length but moving rest frame
does not change relative to another rest frame identically and therefore it seems to expand relative to local rest frame. -i mean if it is even possible to have inertial rest frames that have different unit length (and also time rate due to invariance of light velocity)
hi mfb, about the selection of units- i don't mean the definition of units by using different unit systems, my question is about that the all physics or most of the physics change in reference frame such way that every physical propertys size iow space dimension is K times larger (and also their time rate is K times slower due to light speed invariance principle) than they are in normal reference frame.the change between units between two different sized reference frame together in same place would be
t'=t/K time unit is K times longer
(x',y',z)= 1/K (x,y,z) every space dimension unit is K times larger.
and if the other frame is moving it also has relativistic changes.

and if natural unit length varies from place to place inside same inertial reference frame,
the change between units in different places would be
(x,y,z) = K (x,y,z) and t remains same. also if the frame is moving, it has relativistic effects

 

[Dale]

how does different reference frames interact?

Reference frames don't interact. At least, I can't think of what you might mean by that.

Regarding the rest of your post. For every quantity of interest you could follow the process I did above to determine how it changes under the scale change. I don't really want to work through each one, so I am not going to check if each one is correct or not. It seems like a lot of effort when it is already clear that it is not an inertial frame. But don't just assume that you know how they should transform until you have carefully worked through several in detail.

 

[smm]

thanks for advice, ill look this through

i mean the question how does two particles that are in two different sized frame of reference interact if they collide or have for example electromagnetic interaction.

answer on this Newton 2nd law issue: i think outside forces that act on particles in different sized reference frame has modification factor number in Newton's 2nd law. i am not sure what the factor is- is it 1/K^3 or 1/K or something else

 

[smm]

i think its still inertial reference frame, but it have fixed modification in its kinematics and dynamics.

i try to think that sized inertial reference frames are equal just like different moving reference frames in special relativity are equal - all natures laws are similar for inside observer.

i have idea that reference frame is a physical property that gives also certain different dynamics to particles in that frame.
-Different sized reference frames have different fixed laws of dynamics- similarly than in special relativity the relativistic frame of reference has fixed relativistic properties. but i am not sure what the dynamical properties are.

i don't know would this require that this particle must be isolated from outside world somehow to hold its different properties. i don't know what happens if two different sized reference frames materia mix up together

i think this idea or question is really worth of thinking because the reward is great:
-if there can exist inertial reference frames that can have just any natural unit length, then for example
in theory the whole observable universe can fit into size of one atom, if it has 10^40 times smaller reference frame whose laws of nature follow 10^40 times smaller scale (and 10^40 times faster time due to invariance of light)

 

[smm]

i add to the last "crazy sounding" sentense that if the mini-sized reference frame has also
L^2 = 10^80 times less energy and therefore also 10^80 times less mass the idea of such things comes much more plausable.

but can there be or is it possible in theory to have reference frames in different sizes?

 

[Dale]

i think its still inertial reference frame, but it have fixed modification in its kinematics and dynamics.

This is a self-contradictory claim. If it has any modification in its dynamics then it is not inertial.

i try to think that sized inertial reference frames are equal just like different moving reference frames in special relativity are equal - all natures laws are similar for inside observer.

I proved above that Newton's 2nd law is different.

 

[smm]

quote: mfb

You can just re-label all physical units and corresponding constants. If you do it consistently, you get the same physics again, but Earth has a diameter of 1 meter, light needs 20ns to go around, or whatever. All observers will agree on the physical predictions, and all but you will agree that you use weird units. As you have to change the gravitational constant as well, the Planck units will change their numerical value. You will get the same value for the diameter of earth, measured in Planck units - there is no way to change that (with current physics).


mfb, about the selection of units- i don't mean the definition of units by using different unit systems, my question is about that can the all physics or most of the physics change in reference frame such way that every physical propertys and law have size iow space dimension is K times larger (and also their time rate is K times slower due to light speed invariance principle) than they are in normal reference frame.
-that is- can there be a reference frame at least in theory that follows natures laws in different length scale in all 3 dimensions (and also different time scale due to requirement that light speed is invariant) or shortly : can there be inertial reference frame that has different size or scale.

there are no known phenomenon where this is true, i can think of that at first sight maybe , just maybe ultrarelativistic cosmic rays and hubbles law could involve this, and subatomic radiation (that is in these cases - expansion of reference frame) (-sorry for repetition)

 

[mfb]

If you modify some numerical values of physical constants like the gravitational constant, this is possible. But the result is just a different unit system.

 

[haael]

If I understand the OP well, then he asks if SR exhibits affine symmetry. The answer is: no. You can define an affine transformation of the spacetime, but it is not a symmetry. For instance, it does not preserve the speed of light. Changing scale also means changing energy, mass, temperature and other things. SR is symmetric only under isometric transformations.

You could however change SR slightly, making a variable some parameter which is constant in normal SR that will reflect the changing scale. I wonder if it is not the 4-volume element.

 

[smm]

hi!

in special relativity, particles that are "inside" relativistic reference frame,
has more momentum and also more energy, when they still travel at limited speed near
c - and this energy can be very large - i don't know does it have theoretical limit. this
gives the motivation to my question.

other fact in special relativity what gives motivation to this question is that it doesnt
care what kind of object is "inside" moving inertial reference frame (but the object
must be an isolated system)

i am trying to think this way that the inertial reference frame is a "rigid" object
that covers the space traveler "inside" it - and carries also all physical laws in it -
and observer inside it will observe to be at rest and that all laws of nature are
normal

i could call this by another name, say "state of being at rest" , "rest-being", as i do below, i hope it clarify the issue i am talking about
and my question in this thread is : can there exist "rest-being states" that have
different size or alternatively said a different unit length or unit scale in all 3 dimensions at least in theory, even together at relative rest while this clearly is not known phenomenon in everyday physics

now, if this "rigid state of being at rest" thing i called previously "inertial reference frame" has L times bigger unit length then all laws of nature follow bigger length scale - observer "inside the state of being at rest" still observes all natural laws to be normal.
i mean if rest-being state can have different sizes, or should i say different scales.

clearly - physics and astronomy does not know any "rest-states" that have larger unit
length in all 3 dimensions. for example all atoms in whole universe or those the astronomers can see have similar properties and laws of physics , in other words they are in "same sized rest state" -at least when their relative velocity is near zero.

also there is no known way that for example space traveler could change his "rest-being states" size, in other words, to expand or shrink his "rest-being state"- but according to special relativity he can at least accelerate his relative velocity and then have relative changes in his "rest-being state"

in special relativity knows "rest states" that have different time rate and different lorenz contraction towards the direction of movement and you can draw time-space diagram
and describe the line of simultaenity that changes its angle towards line of light speed,
if the "rest states" relative velocity increases, right

and general relativity describes something to happen to inertial "rest states" what
produces gravitation

but is it possible in theory, while it is not known phenomenon in every day physics and astronomy, to have "rest being state" that has different unit length?

-i already made suggestion that L times larger rest-being state should have L times slower time to avoid situation that its light moves at superluminal speed.
-other suggestion, what is more a guess, is that L times rest-being state would have L^2 times more relative energy if measured from 1 sized "rest being state" - and also relatively
L times smaller rest-being state would have L^2 times less energy than 1-sized state.
(i got this L^2 factor from harmonic oscillators energy 0.5kx^2 - but this factor must be same for all kind of energies)

yes all suggestions makes changes to the rest-being states relative kinematics and
dynamics measured by observer that is in 1-sized rest being state.
-it seems that Newtons I law would hold, (and in fact all constant velocities would be invariant) but Newtons II law would have modification factor
and also Newtons III law would have modification factor when two particles that have
different "rest-being state" interact with each other

but if this "rest-being state" exists, then the observer inside it would still observe
himself being in inertial reference frame... i mean if he measures only physics "inside" his
rest-being state.but question still is open to me- can different sized "rest being states" exist at least
in theory?

 

[smm]

hi! haell thank for the advice! i don't know about mathematics of affinity and other symmetries. have to check about these symmetries more i am not sure what do you mean. the wikipedia seems to have some information about these. i am not talking about selection of units, this is 3rd time i have to clarify this. do i use some wrong word or concept in this thread?

i think this "rest-being state" is a good name of the thing i am thinking here.

the "rest being -state" is state where observer measures all nature laws to be normal
inside this state and also the observer measures to be at rest. this is the same thing, what
"inertial reference frame" is in Newtonian dynamics, right?the thing here below i am chasing is that what kind phenomenon of expansion of reference frame relative to other reference frame could be, how it could happen and does it need energy or not.
(but i don't still know if it is possible to have such reference frames or "rest-being states")

there are no known phenomenon where rest-being state can expand. Here i have some thoughts about modelling how the reference frame or rest-being state
could expand.

but i can make very simple model, where lights rest-being state can expand if the light moves long distance in same reference frame. i do this trick by using the fact that relativistic frame has lorenz contraction. i think this model is simple enough that you can follow it.

if i make following new postulate to special relativity theory:

postulate: -every inertial reference frame has variation in natural unit length such a way that

s' = K(r) s0 where r is distance from origin and :

dK/dr <0
k(0)=1
k(r)>0

that means that natural unit length decreases from 1 to 0 or some value between 0 and 1
when distance from origin increases.

this postulate is totally independent from other postulates in special relativity, isn't it but i don't proof it at any way.
i just make this new postulate to get desired result, (i have heard this phrase somewhere, i don't remember where :) ).

(in fact i could speculate that hubble's law comes from such new phenomenon, then the function K or its linear approximation would be K(r) = (1+H/c)r ) - this is very small number if r < 0.5BLY, something like 1+ 0.75*10^-10 [1/ly]- i have thought this a bit but i don't know about cosmology well enough to say is this right - there are 10 different cosmological tests etc)

this kind of reference frame is no longer eucledian, it looks like a grid observed
through magnifying class. this postulate does not change any nature laws if they are
measured in very small areas.

now when light moves from origin to distant place, its rest being state remains same
because in lights reference frame all distances are zero (here use the lorenz contraction). but local rest frames
or rest-being states natural unit length decreases - and result is that lights rest-being
state expands relative to local rest-being state.

one curious feature in this model is that expansion of reference frame or rest-being demands absolutely no outside energy! that is because nothing happens in the lights own
reference frame during lights journey.

well this is only a model. i think i got the concept of lorenz contraction right.

if the frame expands some how, what ever the phenomenon would be - and the two assumptions below are true:
1. L sized rest-being state has more energy than normal rest-being state
2. during expansion of rest being state the observer inside the state does not
feel any internal forces at all
then the conclusion is that when the rest-being state expands, it gets more energy without
doing any work!

 

[George Jones]

This thread has degenerated into personal speculation, which is against Physics Forums Rules, to which everyone agrees when they register.

I have closed the thread.