Minkowski's 4 dimensional world

[therapeuter]

hello,

i'm reading Einstein's popular book, On the Special and General Relativity,
i'm on Appendix two, Minkowski's Four-Dimensional Space ("World")

http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ap02.htm

however, i ran into such problem here. Einstein says

"We can characterise the Lorentz transformation still more simply if we introduce the imaginary square root of -1 times ct in place of t, as time-variable."

i know that the goal is to square this imaginary square root of -1 times ct in order to get -c^2t^2 which is in the previous simple derivation of Lorentz Transformation:
x'^2 + y'^2 + z'^2 - c2^t'^2 = x^2 + y^2 + z^2 - c^2t^2

but this the imaginary square root of -1 times ct is the distance traveled by light after time t, not time t (time travelled) itself, isn't it? if anyone knows this please tell me.

 

[Garth]

Welcome to these Forums therapeuter!

A good question to start a disucssion.

Einstein notwithstanding, MTW do a good article on this subject: "Farewell to ict" Gravitation Box 2.1, page 51.

This imagainary coordinate was invented to make the geometry of spacetime look formally as little different as possible from the goemetry of Euclidean space.

The problems with it are:

1. Vectors (contravariant) and one-forms (covariant) are confused.

2. Thus it hides the character of the geometric object being dealt with.

3. The essentially very different rotations in Minkowski and Euclidean space are confused.

4. Thus it hides the nature of the parameter in transformations; is it cyclical or does it asymptotically tend to infinity?

5. It hides the completely different metric structure of ++++ and -+++ geometry.

In ++++ Euclidean geometry a zero interval between two events implies they are the same event, in -+++ Minkowskian geometry it implies they both lie on a null geodesic, one event may be a SN Ia explosion at the far side of the universe and the other the observation of that explosion on Earth billions of years later.

6. The causal structure of the universe , limited to all events in the past light cone of a particular event X^a that influence X^a, is broken.

7. Finally, and as a consequence of the above, no-one has been able to discover a way to make an imaginary ict coordinate work in the general curved space-time manifold.

Here, I am afraid, we have to part company with Einstein.

Note also: Stephen Hawking re-introduced ict in his 'theory of everything' to explain what happens to time 'at the north pole', (the BB - "the only boundary condition is there is no boundary condition".) Perhaps that is why he couldn't get that to work either.

Garth

 

[Farsight]

..Here, I am afraid, we have to part company with Einstein.

Note also: Stephen Hawking re-introduced ict in his 'theory of everything' to explain what happens to time 'at the north pole', (the BB - "the only boundary condition is there is no boundary condition".) Perhaps that is why he couldn't get that to work either.

LOL Garth. Excellent.

 

[selfAdjoint]

In spite of all the problems with it that Garth noted and the universal practice of modern physicists, it is worth noting that every pre-WWII physicist and textbook that I've ever seen used ict. It would be an interesting historical study to find out the locus of change.

 

[actionintegral]

Garth,
What do you mean when you say "here we have to part company with einstein"?

 

[selfAdjoint]

Garth,
What do you mean when you say "here we have to part company with einstein"?

Isn't it clear? Einstein's use of ict, as found in his book, does not have the theoretical strength to support Einstein's own curved spacetime in General Relativity consistently, but the modern way of expressing Minkowski geometry does.

 

[therapeuter]

hi Garth, thank you for such detailed explanation...

however, that's not the question i asked...

i was asking, since the - c2^t'^2 in the simple derivation of Lorentz trans. is the minus of the square of the distance traveled by light (r = ct),

http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ap01.htm

then the ict here is the distance traveled by light, right? then why is it that einstein said ict is a substitute for time coordinate?

and if it's distance traveled by light, then how can it make the 4 dimensional continuum analogous with euclidean? for the analogous euclidean equation would simply be:

x^2 + y^2 = z^2,
so x^2 + y^2 - z^2 = 0

right?

 

[masudr]

The Euclidean way to measure distances between two points is to square the differences in all the components, add them, and then square root it. So:

$$ds^2 = dx^2+dy^2+dz^2.$$

We can happily extend this to more dimensions. Say we go for four:

$$ds^2 = dx^2+dy^2+dz^2+d\alpha ^2.$$

Now, as it turns out, in Special Relativity, the only consistent way to measure separations in space time between events is by the following rule

$$ds^2=dx^2 + dy^2+dz^2-c^2dt^2,$$

where all the letters have their usual meanings and units.

So Einstein said, if we make our fourth Euclidean co-ordinate (here written as $\alpha$) equal to $ict$, we'll get (measuring things the Euclidean way):

$$\begin{equation*} \begin{split} ds^2 &= dx^2+dy^2+dz^2+d(ict)^2 \\ &= dx^2 + dy^2+dz^2-c^2dt^2, \end{split} \end{equation*}$$

which gives us the correct way of measuring things.

The motivation behind doing that was so that we could stick to the usual Euclidean distance measuring system (i.e. the Euclidean metric), and not have to change to the new Minkowskian way (i.e. the Minkowski metric). But as it turns out (as Garth has pointed out) the Minkowskian way is actually better for many reasons, and we just have to get used to the fact that the Euclidean metric is not the real metric of SR.

EDIT: it appears I've ignored a large amount of the OP's original post. So I'll try and answer his/her question below...

So the $t$ that appears in the $ict$ is simply the separation in time between two spacetime events. The factor $i$ appears to make the metric Euclidean, and the factor $c$ appears as a conversion factor.

It is really nothing to do with the distance that light (i.e. EM waves) travels in that time t. It is more to do with the fact that nothing (not just light) could travel faster than c, and all massless things travel at c.

 

[Garth]

then the ict here is the distance traveled by light, right? then why is it that einstein said ict is a substitute for time coordinate?

I and others have explained why ict is not the substitute for the time coordinate, the answer to your OP question is simply no:

ict is not the distance traveled by light - that distance is given simply by ct.

Garth

 

[therapeuter]

hi thank you both, i got it. with the help of this page:

http://astro.physics.sc.edu/selfpacedunits/Unit56.html

and i actually didn't notice Einstein discusses this in ch. 26:

http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ch26.htm

i think i might have understood what Garth is saying now...

i guess you are trying to say that the time dimension in Minkowski's 4 dimensional world is just not the same thing as the 4th dimension of a 4 dimensional Euclidean space.

 

[selfAdjoint]

hi thank you both, i got it. with the help of this page:

http://astro.physics.sc.edu/selfpacedunits/Unit56.html

and i actually didn't notice Einstein discusses this in ch. 26:

http://www.marxists.org/reference/archive/einstein/works/1910s/relative/ch26.htm

i think i might have understood what Garth is saying now...

i guess you are trying to say that the time dimension in Minkowski's 4 dimensional world is just not the same thing as the 4th dimension of a 4 dimensional Euclidean space.

That is so very true. The metric tensor of Euclidean space is
$$\left ( \begin {array} {cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right )$$
and the metric tensor for Minkoski spacetime is
$$\left ( \begin {array} {cccc} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right )$$
(or the first element can be 1 and the others -1).

 

[masudr]

Just to add to sA's point, I hope it's obvious that the first component would be $ct$ and the rest are $x,y,z.$

 

[therapeuter]

but einstein's statement that lorentz transformation corresponds to a rotation of the coordinates in 4 dimensional space time is still valid right?

(i also guess that means that motion results in the rotation of your coordinate system: is it right?)

You have to go through the modern linear algebra/differential geometry version of rotation but yes. The Poincare Group is SO(1,3), the orthogonal transformations with unit determinant. Turns out the entries in the matrices can be either hyperbolic functions like sinh and cosh, for boosts, or circular functions like sin and cos for rotations. In Euclidean space you would have its twin SO(4) the four dimensional rotations, with all circular functions. What you lose is Einstein's pretty, but specious derivation of just replacing t with it in the Euler definitions to turn hyperbolic functions into circular functions and Lorentz boosts into rotations.

 

[pmb_phy]

It should be noted here that a space is neither Euclidean nor pseudo-Riemanian until a metric is defined. A metric is something that defines the inner product of two vectors. In SR the metric is pseudo-Riemanian, in particular it is Minkowskian.

What "pseudo-Riemanian/Riemanian" means is a whole different apple.

Pete

 

[therapeuter]

You have to go through the modern linear algebra/differential geometry version of rotation but yes. The Poincare Group is SO(1,3), the orthogonal transformations with unit determinant. Turns out the entries in the matrices can be either hyperbolic functions like sinh and cosh, for boosts, or circular functions like sin and cos for rotations. In Euclidean space you would have its twin SO(4) the four dimensional rotations, with all circular functions. What you lose is Einstein's pretty, but specious derivation of just replacing t with it in the Euler definitions to turn hyperbolic functions into circular functions and Lorentz boosts into rotations.

this seems to mean that euclidean space is a special case of minkowskian space... is it right? (like circle is a special case of ellipsis, when the eccentricity is zero)

 

[MeJennifer]

this seems to mean that euclidean space is a special case of minkowskian space... is it right? (like circle is a special case of ellipsis, when the eccentricity is zero)

No that is not correct.

In Euclidean 4 dimensional space the distance between two points is defined as:

$$ds^2 = dx^2+dy^2+dz^2+d\alpha ^2$$

In 4 dimensional Minkowski space it is defined as:

$$ds^2=dx^2 + dy^2+dz^2-c^2dt^2$$

Or (and I prefer this):

$$ds^2=c^2dt^2 - dx^2 - dy^2 - dz^2$$

The superset of Euclidean and Minkowski space is a 4 dimensional complex space.

 

[therapeuter]

Thanks mejen, what i still don't understand is, what is the difference between

$$ds^2=dx^2 + dy^2+dz^2-c^2dt^2$$

and

$$ds^2=c^2dt^2 - dx^2 - dy^2 - dz^2$$

i mean, if the distance comes up 0.75 using the first equation, it will be -0.75 with the second. is that of no consequence? i mean, like, measuring something forward or backward, no matter?

but then, as i was reading this:

http://astro.physics.sc.edu/selfpacedunits/Unit56.html

"Whenever the displacement in time (cdt) is greater than the displacement in space, the square of the interval is positive. Consider a real body. In the rest frame of the body there is no space displacement, there is only time displacement. The square of the interval is positive." but this is if you use the second equation. using the first you'd have to say, "Whenever the displacement in time (c dt) is greater than the displacement in space, the square of the interval is negative." does it not matter?

 

[robphy]

The difference is called the "signature convention".
http://en.wikipedia.org/wiki/Metric_signature

The first is often referred to as +++-, with the "time coordinate" x₄.

The second is +---, with the "time coordinate" x₀.

A third convention that is used (seen, e.g., in selfAdjoint's post above) is -+++, with the "time coordinate" x₀.

The choice is often one of convenience.

In SR/GR, it makes no physical difference which you use... as long as you are consistent in the application of your definitions and conventions. [It is sometimes a pain to compare the results from different conventions.]

(Similarly, geometric optics has various sign conventions. http://en.wikipedia.org/wiki/Sign_convention)


[Apparently, there is at least one exception to the above:
"Where the sign of the metric makes a difference."
Steven Carlip and Cécile DeWitt-Morette
http://prola.aps.org/abstract/PRL/v60/i16/p1599_1
...but this is beyond the scope of this discussion.]

 

[therapeuter]

excellent clarification, robphy. infinite thanx.

as for mejen, where are you from?

 

[therapeuter]

hi i want to see if i got this right.

i read that the interval between two events connected by light is zero, since the displacement in space, dx, is equal to the time displacement, cdt (x^2 + y^2 + z^2 - c^2t^2 = 0). does this not only explain why time stops for the object that reaches velocity c, but also mean that nothing can go faster than c because spacetime distance can't be negative?

 

[robphy]

hi i want to see if i got this right.

i read that the interval between two events connected by light is zero, since the displacement in space, dx, is equal to the time displacement, cdt (x^2 + y^2 + z^2 - c^2t^2 = 0).

correct.

does this not only explain why time stops for the object that reaches velocity c,

In relativity, no object that wasn't already traveling at c can travel at c. The notion of time for objects that do travel at c is not so clearly defined... have a look at, for example,
https://www.physicsforums.com/showthread.php?t=107741&page=2

but also mean that nothing can go faster than c because spacetime distance can't be negative?

...this needs to be better qualified:
"nothing can go faster than c because the squared spacetime distance along an observer's worldline can't be negative [using the signature convention +---]".

 

[MeJennifer]

Note though that a fast enough spaceship could reach a star system that is 1 light year away in half a year!

While it is true that each frame of reference will see the speed of light traveleing at c, it does not mean that a spaceship cannot go from A to B in any time it wants, except zero time because the ship has mass.

 

[robphy]

Note though that a fast enough spaceship could reach a star system that is 1 light year away in half a year!

Of course, those are measurements from two different frames.

 

[MeJennifer]

Of course, those are measurements from two different frames.

No, only one frame, the frame of the traveler!
His elepased time is real!

 

[jcsd]

No, only one frame, the frame of the traveler!
His elepased time is real!

The distance is from one frame, the time difference is from another.

 

[MeJennifer]

The distance is from one frame, the time difference is from another.

You are right, it is two different frames.

 

[therapeuter]

i have a more serious question now.

after reading that both the mass and kinetic energy of an electron will increase toward infinity when its velocity is approaching c, i realize:

in the frame relative to which it is moving, the electron effectively travels a shorter and shorter distance in the spacetime continuum. this means that the decrease in spacetime distance is compensated by the increase in mass and (kinetic) energy. this means that what is going on here in this relativistic phenomenon is really a conservation process: there is a Total that is necessarily conserved: a Total that is not just mass-energy, but: mass-energy + spacetime distance.

then i read from this article on the equivalence between mass and energy:
http://plato.stanford.edu/entries/equivME/

Interpretations in the second group establish a connection between the terms "mass" and "energy," which are again treated as terms designating properties, and the two basic constituents in the ontology of physics: matter and fields. The equivalence of mass and energy is then taken to show that we can no longer distinguish between matter and fields. Einstein and Infeld (1938) offer a clear articulation of this interpretation. According to Einstein and Infeld, in pre-relativistic physics one can distinguish matter from fields by their properties. Specifically, matter has energy and mass, whereas fields only have energy. Since mass and energy are distinct in pre-relativistic physics, there are physical criteria that allow us to distinguish matter from fields qualitatively. So it is reasonable to adopt an ontology that contains both matter and fields. However, in relativistic physics, the qualitative distinction between matter and fields is lost because of the equivalence of mass and energy. Consequently, Einstein and Infeld argue, the distinction between matter and fields is no longer a qualitative one in relativistic physics. Instead, it is merely a quantitative difference, since "matter is where the concentration of energy is great, field where the concentration of energy is small"(1938, p. 242). Thus, Einstein and Infeld conclude, mass-energy equivalence entails that we should adopt an ontology consisting only of fields.

this means energy is field, which means energy is a spacetime configuration. and the Total that is necessarily conserved in relativistic phenomena is this spacetime configuration that is associated with a measure of energy-mass on the one hand and spacetime distance on the other. any change in one side of this configuration has to be evened out by a reverse change in its other side.

AM I ON THE RIGHT TRACK?

 

[pervect]

this means energy is field, which means energy is a spacetime configuration. and the Total that is necessarily conserved in relativistic phenomena is this spacetime configuration that is associated with a measure of energy-mass on the one hand and spacetime distance on the other. any change in one side of this configuration has to be evened out by a reverse change in its other side.

AM I ON THE RIGHT TRACK?

Nope. More precisely, you seem to be off on your own track. If you want to make your "track" a formal theory, you have to get some physical predictions out of it, and show that they are true. You'll also have to find an appropriate forum, such as the independent research forum. You'll have to develop this "track" quite a bit more before it would qualify to be admitted to the IR forum though.

Meanwhile, the rest of physics is following a completely different track:

1) What is a property of the particle itself is its "invariant mass", which does not change with velocity.

See for instance

http://www.citebase.org/fulltext?format=application/pdf&identifier=oai:arXiv.org:astro-ph/0006423

Does mass change with velocity?

There is sometimes confusion surrounding the subject of mass in relativity. This is because there are two separate uses of the term. Sometimes people say "mass" when they mean "relativistic mass", mr but at other times they say "mass" when they mean "invariant mass", m0. These two meanings are not the same. The invariant mass of a particle is independent of its velocity v, whereas relativistic mass increases with velocity and tends to infinity as the velocity approaches the speed of light c. They can be defined as follows:

mr = E/c2
m0 = sqrt(E2/c4 - p2/c2)

where E is energy, p is momentum and c is the speed of light in a vacuum. The velocity dependent relation between the two is

mr = m0 /sqrt(1 - v2/c2)

Of the two, the definition of invariant mass is much preferred over the definition of relativistic mass. These days, when physicists talk about mass in their research, they always mean invariant mass. The symbol m for invariant mass is used without the subscript 0. Although the idea of relativistic mass is not wrong, it often leads to confusion, and is less useful in advanced applications such as quantum field theory and general relativity. Using the word "mass" unqualified to mean relativistic mass is wrong because the word on its own will usually be taken to mean invariant mass. For example, when physicists quote a value for "the mass of the electron" they mean its invariant mass.

Energy is conserved in GR for "isolated systems" which have an asymptotically flat space-time.


See for instance:

http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Is Energy Conserved in General Relativity?

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

 

[therapeuter]

Nope. More precisely, you seem to be off on your own track. If you want to make your "track" a formal theory, you have to get some physical predictions out of it, and show that they are true. You'll also have to find an appropriate forum, such as the independent research forum. You'll have to develop this "track" quite a bit more before it would qualify to be admitted to the IR forum though.

not sure about this "prediction" and "development" thing. I'm only asking about the philosophical meaning of what i read... to see if it's correct. I'm not trying to make a "formal" theory of the physical world.

but is the "invariant mass" really an established concept? how come i read that since in atomic physics the entire particle can dissolve into radiation, mass and energy equivalence really means that all the mass of a particle can be converted into energy?

but thanks for the input and feedback. i guess the issue is far more complicated than i thought.

does anyone else have any input about the philosophical meaning i was trying to get at?

 

[pervect]

Invariant mass is quite well established, and is an adequate defintion of mass in special relativity (SR).

What you have to watch out for is that the invariant mass of a system in SR is not the sum of the invariant masses of it's components.

Specifically, if you have a particle-anti-particle annhilation, you generate two photons, each of which has an invariant mass of zero. The invariant mass of the system of both photons, however, is NOT equal to the sum of the invariant masses of its component photons, which is zero as each photon has an invariant mass of zero.

Since the energy of a system IS the sum of the energy of all of its components, the best approach is to regard energy and momentum as those quantities that are conserved, and to regard mass as a derived quantity from energy and momentum.

Specificaly, mass is computed from the energy E and momentum p of a system by the equation m = sqrt(E^2 - p^2*c^2) / c^2.

In our photon example, we add together the energy Etotal = E1 + E2 of our two photons. Because the photons are emitted in opposite directions, the total momentum is zero. We can then compute the mass of the pair of photons via the above equation : m = sqrt(Etotal^2)/c^2. This is the equation you alluded to earlier.

The defintion of mass and energy in GR becomes quite technical, unfortunately. From the title of this thread, though, I will assume that you are interested more in SR than in GR.

In SR we can make the following statements:

Space and time can be combined in SR into a geometric quantity, known as a 4-vector. This 4-vector has an invariant "length", which is known as the Lorentz interval. The 4-vector lives in a 4-dimensional "Minkowski" space-time, per the title of this thread.

Energy and momentum can also be combined in SR into a 4-vector. Like the space-time 4-vector, this 4-vector has an invariant "length", which is known as the mass.

As far as conserved quanties go, in SR we have energy and momentum as our primary conserved quantities. Mass can be regarded as something dervied from energy and momentum, a "something" that has the property of being Lorentz-invariant for isolated systems.

There is a deep connection between space-time and energy-momentum. This deep connection is due to Noether's theorem. This is rather advanced stuff, but since you appear to be interested in the philosphical connection between space-time and energy-momentum, according to current accepted theory, I don't see any choice but to mention it.

The symmetry of a system under time translation gives rise to a conserved energy. Similarly, the symmetry of a system under space translation gives rise to a conserved momentum.

See for instance:
http://en.wikipedia.org/wiki/Noether's_theorem

 

[therapeuter]

these are good references. I'm learning something new. especially the Noether theorem. does anyone else have something to say?