Minkowski Space and Space Time

[NonConformist]

can someone please help me to understand these theories or at least provide a site that would help me to better understand them with out getting too technical.

[Cartesiantheater]

can someone please help me to understand these theories or at least provide a site that would help me to better understand them with out getting too technical.

Ha! Finally someone posting an interesting topic!

Where do you want to start?


Minkowski space is a type of geometry in which special relativity is well suited.

Think of Euclidean space first, to help you get an idea (Euclidean space is the normal space of geometry you learned in school). For any point in Euclidean space, you'll have three coordinates (for the three dimensions). Like, for example, Ted might be located at the house on corner of North 14th Ave and Einstein Street (so far we have TWO coordinates), and he might be on the second floor (third coordinate). You know, you have an x axis, a y axis, and a z axis. Any point will have an x coordinate, a y coordinate, and a z coordinate.


Well, spacetime goes further than that. Here is why: any event in the world will have the normal three spatial coordinates (like above), and also a TIME coordinate. If you don't specify the time coordinate, you are not specifying a specific event. You're just specifying a spatial location.

Now, as it pertains to relativity, space and time are very intimately connected in a specific mathematical way (one of the keys being that the speed of light is constant for all inertial reference frames, which leads to some interesting consequences for that mathematical relationship). I can go into more detail if you want, or post some links I might be able to find.



Now on to Minkowski space.

First of all, back to Euclidean geometry to give you an example to compare to the one I'm about to give you for Minkowski space.

You recall the how to calculate distance?

Say you have a point some distance away from the origin (0,0,0) in some Euclidean coordinate axis.

The distance d from the point to the origin is:

d = √(x² + y² + z²)

* if the other point wasn't the origin, (0,0,0), instead of x² it would be (x2 - x1)², and so on for y and z (you've seen this before in geometry, right?)


Okay, back to the equation above. Surely you have no problem squaring both sides, right? So you'd have:

d² = x² + y² + z²

So we can define the space interval like above. It means the same thing as the other way, since distance is always positive.

The reason I brought this up is because there is an analogous thing for special relativity: the SPACETIME interval, which is CRUCIAL for relativity for many reasons, not the least of which is that it is agreed upon by all observers in inertial reference frames. That equation looks like this:

s² = x² + y² + z² -c²t²

You see how it looks very similar to the Euclidean distance formula, except that pesky negative term at the end. The main thing here is that there are three spatial entries and one time entry, and the time entry is negative.



The reason for this is that Minkowski space has what is called a "time-like dimension." (that is another discussion, but you'll probably need to learn about space-like and time-like intervals to have a good grasp of the basic idea).

Notice that every single term has units of m², including the one with the speed of light, c. So it IS a measure of "distance," but not spatial distance. Spacetime distance.





Now, if you want to go further, often times in Minkowski space/ relativity, that last term is considered an IMAGINARY term (i, the square root of -1), and usually that equation is written like this:

s² = x₁² + x₂² + x₃² + x₄²

with x₄ = ict, t is time, c is the speed of light, i is √(-1).




The equation tells us lots of things, including whether or not two events can be causally related (when s² < 0, two events can be causally related, when s² > 0, one event cannot have caused another, and when s² = 0, we are dealing with a photon).

In a nutshell, the whole point of spacetime is to be able to express space and time as similar things, with the same unit (the meter). The reason they ever wanted to do this is because special relativity (because of the way coordinates are transformed between reference frames- i.e. the Lorentz transformation) implies that the way an observer measures time between two events is related to the way another observer measures time between the same to events as a function of distance.

To make this more clear, always look at the math. PRIOR to special relativity, the time measured in one frame would be assumed to relate to the time measured by another frame like this:

t' = t

I.e. both observers measure the SAME EXACT value of time.

But in special relativity, this transformation looks like this:

t' = γ(t - vx/c²)


with γ = 1/√(1 - (v/c)² )

So you see that time is dependent on velocity and distance. That's just the beginning, but it shows that you can't simply take for granted that space and time are forever separate. In special relativity, they definitely AREN'T. They are two sides of the same coin.

So, mathematically, scientists (well, actually the mathematician Hermann Minkowski) wanted to form a mathematical representation of them that showed this. And that is what Minkowski space is all about.

[Cartesiantheater]

Oh, if that is too much algebra for you I'm sure someone here can do a more conceptual heavy explanation. (EDIT- I would really like to help you, but I probably need to know your mathematical level to do so)

Space-time is representing events and coordinates with four dimensions (up-down, left-right, in-out, and time), rather than just the normal three, and Minkowski space is a type of space-time with specific geometrical properties that lend themselves to representing special relativity rather nicely.



If you want to look at it a little more mathematically (I'm sure no one here does), one example would be the standard basis of the space, which is different from, say, the standard basis for R³ for the real numbers (which would look like the identity matrix for R³): the Minkowski one will have a NEGATIVE 1 in the first entry, rather than a positive one, and is four dimensional, rather than three dimensional.

Standard basis of R³ as a matrix

1   0   0

0   1   0

0   0   1

while Minkowski space looks like:

-1   0   0   0

 0   1   0   0

 0   0   1   0

 0   0   0   1

Basically Minkowski space is a type of pseudo-Euclidean space with properties that make it idea for mathematically representing special relativity.

[NonConformist]

Why does it have -1?
Thanks for the explanation, you're the person i was hoping to have answered my question :) Now can you explain to me the way it's referenced in sci-fi, like Star Trek with the whole Space Time continuum?

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. –Hermann Minkowski, 1908 (minkowski space wikipedia page)

could you possibly explain the "independent reality" part

[Cartesiantheater]

Why does it have -1?

In order to do that we'll have to go deeper into special relativity, and this will require some algebra. You okay with that?


Here is the conceptual basis you have to get first: Say you have one guy standing still, and another guy in a smoothly moving train. Say the guy on the train tosses a ball in the direction the train is moving. This is the part you need to understand: The guy on the train will see the ball moving at some speed relative to him, but the guy on the ground will see the ball moving at THAT speed PLUS the speed of the train. Do you understand that?

Well, what has happened is that the speed of the ball in the train reference frame was TRANSFORMED to the coordinates of the guy on the ground- which would require that speed to be added with the speed of the train.


Well, normally, we do this transformation rule (the values with the ' symbol, like x' and t', are the values measured by the guy on the train, and the normal values without the ', x and t, are the values measured by the guy on the ground. Both agree that the train is moving with a velocity u [because that is the relative velocity between the two reference frames]):

called the Galilean Transformation

(displacement) x = x' + ut'

(velocity of the thrown object) v = v' + u

(time rule): t = t'

Likewise, going from the observer on the train you'd have:

x' = x - ut

v' = v - u

t' = t

v is the velocity of the ball, u is the velocity of the train.


Now, special relativity has a different transformation rule: (this is the displacement rule and time rule)

called the Lorentz Transformation

x = 1/√(1 - (u/c)²) *(x + ut)

t = 1/√(1 - (u/c)²) * (t + vx/c²)


And going from the other frame's perspective:

x' = 1/√(1 - (u/c)²) *(x' - ut')

t' = 1/√(1 - (u/c)²) * (t' - vx'/c²)

And finally, the reason for the -1:


Take that spacetime interval and apply it to only one spatial dimension and one time dimension (this makes the math easier):

s² = x² - c²t²

So, the observer in the still frame will have the spacetime interval as:

x² - c²t²

and the observer in the moving frame will have the spacetime interval as:

x'² - c²t'²

(again, with the ')



Now, since the spacetime interval is the SAME regardless of reference frame, then we know that:

x² - c²t² = x'² - c²t'²

even if x ≠ x' and t ≠ t'


So, the key is if you substitute the Lorentz Transformation for each of those values INTO the above equation, you end up getting that they really are equal. * I can work this out for you if you want. It's kind of tedious algebra, but it is pretty neat.

In other words, the -1 is there because it is the exact mathematical relationship that is consistent with the Lorentz transformation, and it is THAT transformation that arises from the postulates of special relativity.

The Lorentz transformation informs us about the nature of spacetime, and from it we get the spacetime interval. (just like with the old transformation, the Galilean transformation, we made the assumption that time was measured the same by any observer- the Galilean TIME transformation it t = t', unlike with the Lorentz transformation)








As far as "where did the Lorentz transformation come from," which I'm sure will be your next question, it comes directly from the hypothesis that the speed of light is the same for all inertial reference frames.

We can go into that detail if you want, but it might seem kind of mathematical (if you're okay with the Pythagorean theorem it shouldn't be bad).

Thanks for the explanation, you're the person i was hoping to have answered my question :) Now can you explain to me the way it's referenced in sci-fi, like Star Trek with the whole Space Time continuum?

A continuum is something that can be forever broken down into smaller and smaller parts- that is, it is continuous, or smooth.

A spacetime continuum is a spacetime that is forever continuous on any arbitrary scale you chose to look at it. (i.e. it is 4-dimensional and smooth)

This really is unimportant except for the more mathematical aspects of the theory (like taking derivatives of arbitrary intervals of the transformation equations), or, more importantly, with General Relativity, in which a huge assumption is that spacetime is entirely continuous (that's why it can be mathematically represented with certain differential geometries).

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. –Hermann Minkowski, 1908 (minkowski space wikipedia page)

could you possibly explain the "independent reality" part

Minkowski here means that the old way (prior to relativity) of thinking as space and time as two truly separate things is simply an inaccurate picture of true reality, and that only a model that unifies them in some way could ever accurately depict nature.

[Cartesiantheater]

Let me restate the above about the -1 factor in a non-mathematical way.

* Minkowski space is a pseudo-Euclidean space that happens to have properties
  that are ideal for depicting special relativity mathematically.

* The reason why there is a -1 factor on the time component is because that
  -1 factor allows for the spacetime interval to be consistent with the Lorentz
  transformation, a mathematical law relating how two observers on different
  inertial reference frames will describe the same event.

* The Lorentz transformation is a transformation law described above with the 
  property that observers in any inertial reference frame will all agree on the
  speed of light regardless of how fast they are moving with respect to each
  other. One of the results of this law is that two inertial observers moving
  at some velocity with respect to each other will NOT necessarily agree
  on how much time has passed between two events or the length of
  objects.

It just so happens that it is mathematically convenient to treat time in Minkowski space as an imaginary distance (i.e. square root of negative 1). There is no other reason other than mathematical convenience, as far as I am aware.


Now, Minkowski's formalism made special relativity a lot more mathematically elegant and useful, and it also shed some light on the fact that special relativity really kind of unifies space and time into a single thing (spacetime), rather than two distinct things like before.

Einstein's genius was realizing that time and space are not absolute, and Minkowski's genius was realizing they are two sides of the same coin (then later Einstein had some more genius with General Relativity, lol).

* Working out the math to show that the spacetime interval defined for Minkowski space IS consistent with the Lorentz transforamtion (i.e. it is invariant under the Lorentz transformation), for the curious (and I'm bored)


Need to show:

s² = x'² - c²t'² = x² - c²t²

So,

x² - c²t² = x'² - c²t'²

The Lorentz transformation says:

x' = γ(x - ut)

t' = γ(t - ux/c²)

and

x = γ(x' + ut')

t = γ(t' + ux'/c²)

Substituting those values in:

x² - c²t² = x'² - c²t'²


(γ(x' + ut'))² - c²(γ(t' + ux'/c²))² = (γ(x - ut))² - c²(γ(t - ux/c²))²


γ²(x' + ut')² - c²γ²(t' + ux'/c²)² = γ²(x - ut))² - c²γ²(t - ux/c²)²

Notice that all the γ² coefficients divide out, which leaves:

(x' + ut')² - c²(t' + ux'/c²)² = (x - ut))² - c²(t - ux/c²)²

Next expand the squared terms:

(x' + ut')² - c²(t' + ux'/c²)² = (x - ut))² - c²(t - ux/c²)²


x'² + 2x'ut' + u²t'² - c² (t'² + 2ux'/c² + u²x'²/c⁴) = x² - 2xut + u²t² - c² (t² - 2ux/c² + u²x²/c⁴)

Now the clever part: realize that x = ct and x' = ct', and substitute those values:

(ct')² + 2ct'ut' + u²t'² - c² (t'² + 2uct'/c² + u²(ct)'²/c⁴) = (ct)² - 2ctut + u²t² - c² (t² - 2uct/c² + u²(ct)²/c⁴)

Distributing everything and simplifying

c²t'² + 2cut'² + u²t'² - c²t'² - 2uct' + u²t'² = c²t² - 2cut² + u²t² - c²t² + 2uct - u²t²

Now combine like terms on the left side (color coded)

c²t'² + 2cut'² + u²t'² - c²t'² - 2uct' - u²t'² = c²t² - 2cut² + u²t² - c²t² + 2uct - u²t²

Adding those left hand terms gives:

0 + 0 + 0 = c²t² - 2cut² + u²t² - c²t² + 2uct - u²t²

Doing the same on the right:

0 = c²t² - 2cut² + u²t² - c²t² +2uct - u²t²

0 = 0 + 0 + 0


Since 0 = 0 is consistent, that means that the spacetime interval is invariant under a Lorentz transformation, by direct proof. □

[NonConformist]

That is just awesome

I liked the train example. If i have anymore questions or something that needs explaining, looks like i know where to ask. thanks again.

[Cartesiantheater]

No need to thank me dude. This is by far my favorite subject to discuss. If anything I should be thanking you. Special relativity threads (that aren't started by me) are few and far between here.

[NonConformist]

Awesome then! lets keep it going. how does Lorentz(sp?) figure into this?
While we're on the subject of explaining things, Paradigms and Paradigms shifts, what gives? Doc you rule, i'm going to pick your brain apart.

[Cartesiantheater]

Awesome then! lets keep it going. how does Lorentz(sp?) figure into this?
While we're on the subject of explaining things, Paradigms and Paradigms shifts, what gives? Doc you rule, i'm going to pick your brain apart.

Well, actually, even though Lorentz figured the math, he didn't apply it nor did he derive it the way Einstein did. Although, originally special relativity was usually referred to as the Einstein-Lorentz theory of special relativity, but that only lasted for a bit, for some reason.


Here's where it starts:

Part I

If you take Maxwell's equations of electricity and magnetism, you arrive at an interesting result: they predict that electromagnetic phenomena are WAVES and that they travel at a peculiar velocity (math worked out below * ). But with respect to WHAT? What medium? Well, the odd thing is that these laws do not say. It just says they propagate at a certain velocity.

Why?

That was the question 19th century physicists wanted to answer. The best explanation anyone could figure out was that these waves travel with respect to an all pervasive medium called the luminiferous aether.


A great hypothesis, but useless if it can't be shown to actually exist.

So, on to part II.

Part II

Scientists wanted to measure the luminiferous aether to lend some physical evidence to the theory. So they set out to measure it. How would you measure this? Well, if planet Earth was moving through the aether, that means that the velocity (which we knew) that we measure of light should depend on the direction that we shine it, because there should be a "current" or "aether wind" as the earth travels through the medium.

The only problem was that no one could measure a difference regardless of which direction the light shined. This was NOT because we lacked the ability to measure such velocities. Why? Lorentz to the rescue.

Part III

Lorentz argued that the atoms in the devices we used in the apparatus used above were "squeezed" together by the aether current in exactly the right amount that would forever prevent us from seeing the distinction in the velocity of light. Therefore, the idea is that the aether itself interfered with our ability to measure these differences.

Einstein, however, had a different idea.

Part IV

Einstein suggested that instead of an aether screwing with our measuring abilities, the problem was that we were taking the nature of space and time for granted. Einstein said that there WASN'T something fishy going on with our measurements. We really were seeing that the velocity of light was independent of our motion through the aether, which to him suggested that there WAS no aether. Instead, he argued that the way we transform velocities from one inertial reference frame to another was fundamentally WRONG because it's assumptions about the nature of simultaneity, that is space and time, was utterly mistaken.

He took Lorentz equation describing Lorentz' hypothetical "contraction" etc, and suggested that rather than describing some effect the aether was having on objects, it instead described a fundamental aspect of space and time itself.

So, Einstein made two assumptions about the universe, and proceeded to derive Lorentz' equations from them.

Part V

Where the equations come from?


Assumption (1): The laws of physics are the same for all inertial reference frames.

Assumption (2): The speed of light is the same for all inertial reference frames.




The first one is vital, but to understand what's going on it's better to pay attention to the latter. This all depends on simultaneous events.



What are simultaneous events? Two events separated in space that according to an observer at rest with respect to them occur at exactly the same time. Here is how Einstein did it:

Assume you are sitting directly in the middle of two mirrors separated by some distance. If you turn on a light bulb, the light will move to the mirrors and then back to you. If you see the reflected light from both mirrors hit you at the same time, then the point where the light hit each mirror is defined to be simultaneous.

Now, imagine if there is an observer moving parallel to the line that the light travels at some velocity. Does he agree that the two reflected beams of light hit you at the same time?

If both that observer and YOU measure the same velocity of light in all cases, the answer to that question is NO- he does NOT agree with you that the two reflected beams get to you simultaneously.


Why? This is pretty easy to visualize with some animations.

Part VI

The first one is two light clocks at rest with respect to each other. A light beam travels from one mirror and back down to another. The light travels at exactly the same speed for both of them.




Now, what happens if one of these light clocks moves at some velocity with respect to the other? (remember, the LIGHT ITSELF moves with the same velocity in both cases).




As you can see, in the moving light clock, the photon actually travels a greater distance. This means that it takes longer for the light to go from mirror A to mirror B.


Why does this matter? Simple.


According to the principle of relativity (not the special one; this is pre-relativity), there is no distinction between defining an object as moving with respect to you and defining the object at rest and YOU are the one moving. (for example, if you're in a smooth moving bus, you are mathematically justified in saying that you are at rest and all the people zipping by are the ones in motion- for this very same reason we can't really say either the earth is moving and the stars are at rest, or the stars are moving and the earth is at rest, etc. It's meaningless to attach any definite meaning to either case).

This means that a MOVING OBSERVER looking at a "stationary light clock" would see that the light takes longer to get from mirror A to mirror B than an observer who is stationary with respect to the light clock.


Here is the important part: If both observers agree on the speed of light, then why is there a difference? The difference must come about because of how the two observers measure time and space- they must measure the time and distance of the moving light beam to be DIFFERENT, otherwise they would both agree that the photon takes exactly the same amount of time to traverse the distance. And they DON'T which means they must disagree on measurements of space and time.


For this reason, whether or not two events separated by a certain distance are simultaneous or not depends upon your frame of reference. YOU see that the events are simultaneous, but Observer A does not.





It just so happens that if you assume that the speed of light is the same in both reference frames, you can (very easily) derive Lorentz' equation from it using more or less the Pythagorean theorem.

Lorentz' equation essentially quantifies the differences that two observers will see between how they measure the spatial or time interval between two events.


A quick derivation of the time relation (using the light clock example):

** note: if you understand the above you don't need to see the math to be convinced, but I understand it is very tricky. When I first learned this the math was a crutch I used that helped me understand it. Of course, if you're not familiar with elementary algebra, it isn't going to help to much, lol!

From the perspective of the person at rest with respect to the clock, the light travels a distance c∆t. From the perspective of the watching the clock move, the clock moves v∆t' distance, and the light travels c∆t' distance (remember, distance = rate X time. For light, we assumed that the rate is c, the speed of light, for all observers. ∆t means "change in time" for the non-moving reference frame, and ∆t' means "change in time" for the moving reference frame. v means the velocity with which the moving frame is moving).





Then, using the Pythagorean theorem:

(c∆t')² = (v∆t')² + (c∆t)²

c²∆t'² = v²∆t'² + c²∆t²

Divide by c²

∆t'² = (v²/c²) ∆t'² + ∆t²

Then solve for ∆t'

∆t'² - (v²/c²)∆t'² = ∆t²

∆t'² (1 - v²/c²) = ∆t²

∆t'² = ∆t²/(1 - v²/c²)

∆t' = ∆t/√(1 - v²/c²)

And there is the time dilation relation.


Now divide both sides by ∆t and you have the Lorentz factor. If you just multiply this factor by the Galilean transformation laws, you have the transformation laws for special relativity (this works because they are linear transformations. This DOES NOT work for the special relativity VELOCITY transformation, because those equations are NOT linear. Definition of linear in this context).

Hopefully that gives you a quick run down of the history. Remember that Minkowski space is a type of space that is consistent with the Lorentz transformation equations, which are (partially) described above. Now you know where those equations come from, so that should more or less complete the question. Of course, you could always go into more detail, lol, but you now have the logical basis for special relativity in a nutshell.

... but as you no doubt are aware, it's the RESULTS of the the theory that are the most interesting ;)

* Math showing Maxwell's equations lead to a wave equation (of course, this is only for consistency in my post, since I'm sure about 1% of the posters here can follow- it's no big deal, vector calculus is hard):




If you start with the Maxwell-Faraday equation

▼ X E = - ∂B/∂t

(1) Take the curl of both sides, noting the vector identity

▼ X (▼ X A) = ▼(▼• A) - ▼²(A)

gives

▼(▼• E) - ▼²(E) = - ∂(▼X B)/∂t

but remember that

(2)

▼ X B = με ∂E/∂t

(this is Ampère's circuital law)

Where μ is the permeability of free space = 4 π x 10^-7 T m/A

and ε = permittivity of free space = 8.854 x 10 ^-12 C^2/Nm^2

(the two constants I mentioned)

So substitute Ampère's circuital law for the curl of the magnetic field B, giving:

(3)

▼(▼• E) - ▼²(E) = - με ∂(∂E/∂t)/∂t

which is

▼(▼• E) - ▼²(E) = - με ∂²E/∂t²

And then remember that

(4)

▼• E = 0 when there is no charge

(this is Maxwell's equation for an electric field in a vacuum)

So, now you have:

(5)

- ▼²(E) = - με ∂²E/∂t²

so dropping both negative signs gives:

▼²(E) = με ∂²E/∂t²

Which if you can see, that is in the form of a wave equation- here is a standard wave equation so you can see that it is in fact a wave equation:

(6)

∂²y/∂x² = (1/v²) ∂²y/∂t²

But in this case, the portion of the wave equation that contains velocity v is με, so if you equate the two corresponding portions of the two corresponding wave equations and then solve for v,

(7)

(1/v²) = με

v² = 1/με

v = 1/√(με)

And then substitute in the numerical values for those constants,

(8)

1/√(με) = 1/1/√(4 π x 10^-7 T m/A * 8.854 x 10 ^-12 C²/Nm²)

1/√(με) = 2.998 X 10^8 m/s

^^ that just happens to be the speed of light.

[Cartesiantheater]

Something I want to add on this:

Vectors:


Normally we think of a vector as a thing that has a direction and a magnitude. In three dimensional space, one way to write a vector is like this (the 3-Dimensional vector A):

A = [x y z]

and this means that if you put the vector's "tail" at the origin, then the "head" of the vector would be at the point (x, y, z).
So a vector whose "head" is (2, 3, 5 ) would look like this:


(the red one is the vector)




Well, Minkowski space also has vectors, called 4-Vectors. They are vectors through spacetime, so they have three coordinates for space and one for time.

One other major difference is that the distance formula is slightly different. Minkowski space is hyperbolic in nature, and so "distance" (actually these are spacetime intervals between two events in Minkowski space) will always have a negative sign on either the time or spatial coordinates.

Distance in Euclidean space (regular every day geometry- you've seen this for two dimensions in the Pythagorean theorem for right triangles in school, the old a² + b² = c²):

s² = x² + y² + z²

Spacetime intervals in Minkowski space (geometry of special relativity):

EITHER

s² = t² - x² - y² - z²

OR

s² = -t² + x² + y² + z²

(* Why it can be either has some to do with the fact that spacetime intervals can either be time-like or space-like, with the order of events being preserved in time-like intervals (first one), whereas in space-like intervals events are two far away to be causally connected, which means that the order of events is relative. The only requirement is that if you chose to represent something using one particular distance formula, you MUST be consistent with it. As far as I know, the preference for which is better is still debated)

More information here:
http://en.wikipedia.org/wiki/Sign_co...tric_signature

Anyway, the other important thing about 4-vectors is that they are consistent with Lorentz transformations, which by now we all know is the heart of special relativity.



That pretty much wraps up the basics of it. If you then apply this stuff about Minkowski space to physics, like velocity, momentum, and energy, you'll find some neat stuff. Just be careful you don't forget that time and space are relative, and only proper time and proper length are invariant- proper time being time measured by a clock that is at rest with respect to an observer, and proper length being a length whose ends are measured simultaneously with respect to an observer. So if you want to look at a vector in Minkowski space, be sure to use proper time.