I made a dimensional analysis of Einstein field equation, and I found L=T (space dimension= time dimension). Do you have an explanation for that?
Article Dimensional Analysis of Einstein's Fields Equations
In school mechanics we consider three basic units, time, length and mass. Units of all other quantities are written in terms of these basic units. Then, as new quantities are introduced (like electromagnetic fields) we need to introduce new basic units (like charge).
But there are some basic physical constants like velocity of light, Plank's constant and Newton's gravitational constant. Because they are fundamental constants, it is often convenient to choose system of units in which the values of these constants are chosen to be unity. In these system of units, also called natural system of units, time and length have same dimensions. You can express time in units of length or vice versa. When one wants to convert a quantity from natural units to units used in everyday life, one needs to multiply by appropriate factors of natural constants.
So, in an equation length and time will have same dimensions and may be expressed in units of length or time. From length one goes to time by dividing by velocity of light. Similarly, from time units one goes to length unit by multiplying by velocity of light.
This is what Robert Low has called geometrized units but there is no geometry involved here. It is a matter of choosing natural system of units. It is natural system because natural constants are chosen to be unity.
ok. but without taking c to be dimensionless? I mean: I did my dimensional analysis with the raw Einstein's field equations (not making any normalization of time and space and of c).
you say: 'it is conventional', is there a reason?
Einstein's equation states that the left side (geometry) is equal to the right side (energy). By doing that you are implicitly choosing the dimensions of the quantities entering, i.e. the convention that c is dimensionless and equal to1.
There are lots of examples of physical laws stating that A is proportional to B. If A and B have different dimensions, the law must contain a proportionality constant. An example is Newtons law of gravitation, where the gravitational constant G has the necessary dimensions.
Thus if you don't choose c=1 dimensionless, GR must be written A= QB, where the proportionality constant Q is only introduced to make A and QB have the same dimension.
General Relativity theory takes for granted that the speed of light in vacuum (usually denoted by 'c') is a universal constant which does not depend on the state of motion of the observer with respect to which such speed is considered. But its value may vary according with units chosen, being approximately equal to 3.10^8 meters/second or 6.10^13 miles/year or others as you please. An appropriate choice of units of length and duration can make this constant to be numerically equal to one (c=1). Using such units, the standard relation L=c.T reads L=T as you wrote it down. The physical content of this is indeed to measure the duration of a small lapse of time as being equal to the length of the trajectory of light propagating in vacuum along that lapse of time.
@Robert: I took the metric tensor to be dimensionless in order for the left hand side of the equation to be homogeneous. Unhappily I do not have that book. I will look for it.
ok, I agree that it makes life simple but maybe time has really a space dimension (or at least is a function of a space characteristic dimension).
@Matts: taking c dimensionless does not change the result of the dimensional analysis. could you give an example of this proportionality constant Q? I did not read about that...
to conclude: Einstein's field equations (EFE) are not homogeneous and one can see that in Friedman's equations which are an exact solution of EFE
Nathalie,
If a galaxy of mass m is located at a radius r from the center of a sphere of mass M it will acquire a radial acceleration (Newton's law) a=-GM / r^2. Time enters only on the left side since the acceleration is the second time derivative of r. This relation only expresses proportionality unless you endow the constant G with suitable dimensions, such as m^3 /(kg s^2),
Usually on writes the law as the force F exerted by the mass M on m.
F = -GMm / r^2
You can express any of the measurable quantities in this law in whatever units you like, in the end the proportionality constant G has to carry suitable dimensions.
Robert,
I should have added that G also enters in Einstein's equation, which states that the left side is proportional to the right side, and it is up to you to endow the constant G with suitable dimensions.
Wait, Robert! if you use non geometrized units, the stress energy tensor has dimensions of an energy density but also divided by c^2 (relativistic mass), so the dimensions of the stress energy tensor has dimensions MT^{-2}L^{-1}L^{-2}T^2=ML^{-3}...
then finally T=L and the EFE are not homogeneous...
Matts, I agree that G should have suitable dimensions. I only question the usual (conventional way) EFE are treated.
@Renato: sorry I only saw your post now: you make a confusion between units and dimension, these are not the same.
Robert: I refer to the beginning of the discussion, when you were talking about c=1...
Everyone seems to use geometrized EFE...
Robert: as Matts says, one has only to arrange the dimensions of G...
By the way: are the metric tensor and the stress energy tensor homogeneous?
i.e. can we give them a unique dimension (for each component of the tensor and in all cases of GR)?
Nathalie, look at arXiv:astro-ph/9608106, arXiv:gr-qc/9510060, arXiv:1106.1082 which appear to give useful answers to your question. To my understanding the answer is no, not in all cases of GR.
Robert, my question was version 1 and your answer is interesting!
Another question: to make numerical simulations to compute the results of the EFE (like for a GPS system, or in astrophysics and cosmology), do people use geometrized units or non geometrized units? in the first case, one may lose the real numerical values and in the second case, part of the EFE may not be homogeneous (?).
I really do not know anything about numerical computations in GR so it is maybe a naive question.
In school mechanics we consider three basic units, time, length and mass. Units of all other quantities are written in terms of these basic units. Then, as new quantities are introduced (like electromagnetic fields) we need to introduce new basic units (like charge).
But there are some basic physical constants like velocity of light, Plank's constant and Newton's gravitational constant. Because they are fundamental constants, it is often convenient to choose system of units in which the values of these constants are chosen to be unity. In these system of units, also called natural system of units, time and length have same dimensions. You can express time in units of length or vice versa. When one wants to convert a quantity from natural units to units used in everyday life, one needs to multiply by appropriate factors of natural constants.
So, in an equation length and time will have same dimensions and may be expressed in units of length or time. From length one goes to time by dividing by velocity of light. Similarly, from time units one goes to length unit by multiplying by velocity of light.
This is what Robert Low has called geometrized units but there is no geometry involved here. It is a matter of choosing natural system of units. It is natural system because natural constants are chosen to be unity.
Please read the two page paper:
http://www.e-physics.eu/SpacetimeVersusPaginatedModel.pdf
Time and space are, of course, different concepts. Equalization of them is a typical feature of relativistic physics. It was an important heuristic step, it formulated viewpoint that enables to understant many features of physical reality. However, the view from any specific viewpoint is only partial view, some aspect it makes clear, other aspect unclear. Till today we can see nature from several viewpoints (relativistic, quantum, psychological ...) and even the physical viewpoinst are not unified. There are many contradictions between them.
Most importat difference between space and time is flow of time. And this aspect is not reflected by relativistic approaches. Is it "only psychological" aspect? (Plato, Plotinus..."time being the moving image".) I thing, that it is the weakest part of relativity and also the strongest chalenge for future development of physics....
Peter,
Special relativity is based on the requirement of Lorentz symmetry. But that requirement is irrelevant to human observations, to the psychological aspect of life where time and space appear to be "different concepts". Nothing we observe moves near the velocity of light; yes, light does, but that we don't experience. Neither did Plato nor Plotinus, so let's not judge relativity by human observations..
The same goes for GR. It is explicitly constructed to coincide with classical Newtonian physics in the "weak field limit". It is not in conflict with human experience because we live in the weak field limit.
In special relativity too time and space are two different things. It is a statement regarding relationship between two coordinate frames moving relative to each other. Based on behavior of light, it finds that flow of time is at different rates in two frames and there is a relation between distances and time in one frame with those in another. In both frames time flows in one 'direction' in all frames but rate appears to be different when seen from another frame. Relativity does not really equate space and time. For example, if two events are simultaneous in one frame, they may not be simultaneous in another frame but one can never make them appearing in same space point. Technically speaking, the quantity t*t - x*x is same in all inertial frames. If t is zero in one frame, this quantity is negative and one cannot make it positive in any frame. Same is true of general relativity.
We have nonrelativistic physics when velocities of objects are much smaller than velocity of light. All relativistic physics goes over to nonrelativistic physics in this limit. So, relativistic physics is not something special. non-relativistic physics is, on the other hand, a special case. Same about quantum mechanics. Classical mechanics is a special case of quantum mechanics. I do not understand what is psychological physics(?). Haven't come across it so far.
In a spacetime model the common notion of time is coordinate time, which ticks at the location of the observer. It is a mixed concept because it involves that spatial trajectory that information takes to travel from the location of the observed item to the location of the observer. This selection of the notion of time mixes space and time.
Proper time ticks at the location of the observed item. It is independent of space and it is a pure measure of progression. The problem of proper time is that in general proper time cannot be measured. It can only act in deduced space progression models. On the other hand these models are simpler than the spacetime model that is applied by contemporary physics.
Special relativity is defined in terms of space and coordinate time.
The surprise that was sprung by special relativity (SR) that anything that moves with speed of about 3 times 10^{10} cm/s, call it c, (it so happens that only zero rest mass particles can and necessarily move with speed c as ordained by SR) will be measured to have the same speed c no matter which reference frame one is in, leads to other counter-intuitive notions like simultaneity of events not being absolute, etc.
Therefore, c being a fundamental constant can be used to convert time intervals into spatial distances.
But one inherent difference between time and space is that time must flow and can flow only in one direction. This is an enigma.
"Therefore, c being a fundamental constant can be used to convert time intervals into spatial distances." holds for units and not for notions of space and time.
Hi Hans,
At the same time (no pun intended), when one goes from one inertial frame to another, time and space get mixed up (due to Lorentz transformation). Hence, time can be viewed as a fourth dimension (as emphasized by Minkowski).
Contemporary physics uses a spacetime model in which space and coordinate time are coupled via the local speed of information transfer. Coordinate time ticks at the location of the observer.
Other space-progression models are possible that still are compatible with the spacetime model.
An example is the paginated model in which all proper time clocks are synchronized. Proper time ticks at the location of the observed object. In this model universe proceeds with universe wide steps from one static status quo to the next static status quo. It means that in this model space and time are completely decoupled. An external mechanism controls the recreation of the complete universe at every progression step.
In the paginated model time can be considered as a fourth dimension, but is is quite distinct from the spatial dimensions and cannot be exchanged or mixed with one of these spatial dimensions.
If the curvature of space is known, then in principle the spacetime model can be converted into the corresponding paginated model.
Contemporary physics uses a spacetime model in which space and coordinate time are coupled via the local speed of information transfer. Coordinate time ticks at the location of the observer. The spacetime model has a Minkowski signature.
Other space-progression models are possible that still are compatible with the spacetime model.
An example is the paginated model in which all proper time clocks are synchronized. Proper time ticks at the location of the observed object. In this model universe proceeds with universe wide steps from one static status quo to the next static status quo. It means that in this model space and time are completely decoupled. An external mechanism controls the recreation of the complete universe at every progression step.
In the paginated model time can be considered as a fourth dimension, but is is quite distinct from the spatial dimensions and cannot be exchanged or mixed with one of these spatial dimensions.
The paginated model has a Euclidean signature.
If the curvature of space is known, then in principle the spacetime model can be converted into the corresponding paginated model.
If c=1, the dimension of time is the same as length. However, there is a difference between time and length, expressed in the metric. In flat space, the usual choice of signature is (-+++), where the first sign corresponds to the time-time component of the metric. In curved space a known analytic solution is the Schwarzschild case. Ouside the black hole, the signature of g_(00) is negative and the one of the radial component is positive. Inside the black hole the signature is interchanged, thus, radial and time component interchange their meaning. Time is now a length and r is time.
@ Peter Hess:
In flat space one could choose the metric (+ - - - ) and do everything without any problem. The fact that time increases monotonically is still preserved.
My question is, inside the black hole, how does the change in the metric affect this fact. Does it mean space is increasing monotonically and time can be reversed? But then one can always construct a locally inertial frame in which space is flat over a small enough region of space. What happens in this small region of space? I suppose according to equivalence principle, physics should be same as that in free space in absence of gravity and therefore time would be increasing monotonically.
What do experts say?
Due to its event horizon, we do not know anything about the internals of a black hole.
Everything that is stated about those internals depends on the used model. Even the event horizon itself is part of a model.
It could be so that everything that is inside of the BH is packed inside the surface of a sphere that coincides with the event horizon. It would go nicely with the holographic principle and with Bekenstein's findings.
A paginated model is far simpler comprehensible than a spacetime model. You do not have to struggle with the Minkowski conversion that is described by a Lorentz transformation. You must only have some trust in your knowledge of the notion of proper time.
Once again we run into the dichotomy between the inner and outer worlds. Yes, Einstein and Minkowski & co. were right in a sense to objectively set time on a par with space. But our inner, virtual reality space of consciousness experiences the qualia of time in a very special way - in that sense time is also Husserl's prehension -
(sorry broke off too soon) - and Whitehead's actual entities etc. As with all subjective qualia, the time sense is ineffable. As Hawking says, it's all very well to have the equations with time and space - but “What is it that breathes fire into the equations and makes a universe for them to describe?”
The safest view is a paginated model. Without an external correlation mechanism in a paginated model space and progression are completely decoupled. In the paginated model proper time is used. It is a pure notion of progression.
In a space time model coordinate time is used. This is a mixed notion of both space and progression.
Dear Shashikant,
>….In school mechanics we consider three basic units, time, length and mass….…So, in an equation length and time will have same dimensions and may be expressed in units of length or time…
I suggest you have a look at my paper "The Geometrical Meaning of Time" everything is explained there in detail. (http://dx.doi.org/10.1007/s10701-008-9215-3)
There are also Los-Alamos archives versions:
gr-qc/0602034, gr-qc/0611124
@Asher
If you switch to an Euclidean space-progression model by using proper time instead of coordinate time and then synchronize all proper time clocks, then you arrive at a paginated model
In curved space models proper time cannot be measured (without knowing curvature)
All space-progression models that consider the notion of an observer do possess the notions of proper time and coordinate time.
The coordinate time clock ticks at the location of the observer.
The proper time clock ticks at the location of the observed item.
Special relativity is formulated in terms of coordinate time and space.
Coordinate time and proper time differ due to the fact that information needs to travel from the location of the observed item to the location of the observer.
A paginated space-progression model is special because in that model all proper time clocks are per definition synchronized.
In a spacetime model the proper time clocks do not need to be synchronized, but it can be done. In that case the spacetime model and the paginated model are different views of the same reality.
The full form of Einstein's equations, in SI units, is
R_{ab} - (1/2) R g_{ab} = (8 Pi G / c^4) T_{ab},
where G is Newtonian gravitational constant. Try the dimensional analysis of this form of Einstein's equations!
Shashikant Phatak and Matts Roos explained the problem correctly: your problem is merely a problem of units. I will not repeat their argumentation, which is correct, but let me use the example. It is absurd to measure cosmological distance in meters. One of the most convenient measure of distance is the light year, familiar to everybody. It is called "light year" which suggests that it is a time, but that's not true: light year is a distance. It is a distance which light absolves during one year.
In physics we often use the "light meters". Again, the light meter is not a distance, it is a time which the light needs to travel along the distance 1 meter. In this units, the speed of light is
c = (1 meter) / (1 light meter) = 1.
Then the velocities are dimensionless and by velocity v we actually mean velocity v*c. This is the source of your problem. In these units, the lengths and time have the same units.
Robert, I agree that time is a special dimension within geometry.
@Nathalie
There is nothing to agree or disagree with the FACT that in pseudoriemaniann spacetimes there is a difference between timelike direction and the spacelike directions :) But what about the issue of dimensions? Is it clear now?
As far as DIMENSIONS are concerned, if you take c=1, i.e., the local speed of light in vacuum equal to one, it is correct to assign the same DIMENSIONS,
@Martin, The problem in physics is that everybody uses mathematical models which fit experimental phenomena and then they call the model a FACT. With more than 3 parameters you can fit whatever equation to whatever experimental data by changing the values of the parameters...I intute that there will be other models which fit better 'spacetime' than pseudoriemanian models in the forthcoming years..
Of course it is an exaggeration!
What I mean is that physicists should have a 'physical sense'.
I don't understand your example? In particle physics, they make the mathematical model first (except the SM) and then they try to fit experiments to their models. It's worse than trying to fit a mathematical model to experimental data (I'm exaggerating here again).
Do physics not maths...
'It's worse than trying to fit a mathematical model to experimental data' oops!
Yes,
but also take into account that the time has only one direction in our reality.
There is an old article by A.Saharov Cosmological models of the Universe with reversal of time’s arrow
http://ufn.ru/en/articles/1991/5/k/
>....
but also take into account that the time has only one direction in our reality...
Baurov Yu.A. We can discuss this problem.:Advance a hypothesis 1. Let the observable three-dimensional space R3 be formed as a result of minimization of potential energy of byuon VSs interaction in the one-dimensional space R1 formed by them. More precisely, the space R3 is fixed by us in consequence of dynamics of objects generated by interaction of VSs of the byuon. That is, dynamic processes arise in the space R3 for objects with residual (after minimization) positive potential energy of interaction of byuon VSs, and, as a result, wave properties of appearing elementary particles come into existence.
Please see more :1. Baurov Yu.A., On the structure of physical vacuum and a new interaction in Nature (Theory, Experiment and Applications), Nova Science, NY, 2000.
2. Baurov Yu.A, Global Anisotropy of Physical Space. Experimental and Theoretical Basis. Nova Science, NY, 2004.
The answer on "Is time equivalent to a space dimension?" depends on the applied model. Since this is a rather fundamental concept, the model must contain a foundation from which space and progression emerges.
Be aware contemporary physics applies quite different notions of time. Part of them are not pure versions of progression. Instead they are mixtures in which also space plays a role. Proper time comes close to progression. In my private model all proper time clocks are synchronized.
Dear Prof. Hans van Leunen Please see my paper in Open Journal of Microphysics, 2011, 1, 35-39
"Byuons, Quantum Information Channel, Consciousness and Universe"
The Quantum Information Channel connects all objects of nature. If time clocks are synchronized in your model we can have a some connection (If it is't Newton)
Dear Prof.Glenn Borchardt: time is a seguence of events. What ?Please see Open Journal of Microphysics, 2011, 1, 35-39
Preprint Equivalent Fundamental Physical Quantities and dimensions in...
Equivalent Fundamental Physical Quantities and dimensions in special relativity
I want to say the
Gravitational constant = m3⋅kg−1⋅s−2
not constant
Preprint Do the Planck length, time and mass follow the Lorentz contraction?
No it is the dimension of the real axis of a quaternionic number system. The other three axes are spatial axes.
Dear
Nathalie Olivi-Tran,
What is the nature of time? Answer of the question is- Receiving imagination in the brain of living beings named primitive mankind, it became detached from other living beings. In our home planet so many life yet, various imagination power of brain is only for the mankind. By the Little Imagination power of Primitive mankind make of the first policy the knowledge of awareness of time. The day after night and in the daylight, they proceed to the development work of the primitive beast mankind to find out the way to live well. Hence, time is not from the origin of the Transcendent/Infinite. The knowledge of awareness of time is just result of imagination from the primitive mankind. Consequently lawfully, we can take the resolution that- Idea of time is limited within the respective location of every one, idea of time from the outside of the location of oneself is valueless and the locations of oneself are the present. So, reality of anything shall not be received for anything except the respective places of each in the universe. In the Universe there is no separate time dimension but the very calculation of arrival or rolling and moving is the assessment of value of modern time. Hence, people can take the resolution that-
Is time no equivalent to a space dimension.
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I am looking forward for your response
The hyperbolic Lorentz transform converts Euclidean combinations of proper time and three dimensional space locations into spacetime coordinates by inserting time interval dilations and length contractions that depend on the relative speed of the observed event and the observer. All observers perceive in spacetime coordinates and can only perceive events that occur in the past. The embedding field (our universe) transports the information and gets deformed by massive objects in the neighbourhood of the information path.
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