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

1. Dec 14, 2012

### 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.

2. Dec 14, 2012

### Mentz114

3. Dec 14, 2012

### Staff: Mentor

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

4. Dec 16, 2012

### smm

hi!

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 lets 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 theorys 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 dont know enough about astronomy to think this trough - there i heard is 10 cosmological tests that can test different models of space.

Last edited: Dec 16, 2012
5. Dec 16, 2012

### 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

6. Dec 16, 2012

### bcrowell

Staff Emeritus
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

7. Dec 16, 2012

### 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?

Last edited: Dec 16, 2012
8. Dec 16, 2012

### HallsofIvy

Staff Emeritus
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.

9. Dec 16, 2012

### smm

hi! no, i dont 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 dont 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?

Last edited: Dec 16, 2012
10. Dec 16, 2012

### Staff: Mentor

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

11. Dec 16, 2012

### bcrowell

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

12. Dec 16, 2012

### smm

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

well i dont 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??

Last edited: Dec 16, 2012
13. Dec 16, 2012

### 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.)

14. Dec 17, 2012

### smm

sorry about my english, it may be that i may use some wrong words and wrong name- i dont 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 dont 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)

Last edited: Dec 17, 2012
15. Dec 17, 2012

### Staff: Mentor

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?

16. Dec 17, 2012

### 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 cant 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 cant 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.

Last edited: Dec 17, 2012
17. Dec 17, 2012

### Staff: Mentor

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).

18. Dec 17, 2012

### Staff: Mentor

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:
$$m\frac{d^2 X}{dT^2} = m\frac{d^2(kx)}{d(kt)^2} = m\frac{k}{k^2} \frac{d^2 x}{dt^2} = \frac{1}{k} f$$

Last edited: Dec 17, 2012
19. Dec 17, 2012

### 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".

-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 dont 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.

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 dont 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 dont 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

Last edited: Dec 17, 2012
20. Dec 17, 2012

### Staff: Mentor

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.