Units of spacetime in Minkowski metric

[nomadreid]

In the equation
ds²=dx²+dy²+dz²-c²*dt²
the units on the RHS are units of distance squared. But it would seem that units for a spacetime metric should somehow be in units which incorporate both space and time units.
Undoubtedly this is an elementary question, but one has to start somewhere. Thanks.

 

[ghwellsjr]

In the equation
ds²=dx²+dy²+dz²-c*dt²
the units on the RHS are units of distance. But it would seem that units for a spacetime metric should somehow be in units which incorporate both space and time units.
Undoubtedly this is an elementary question, but one has to start somewhere. Thanks.

That's actually the formula for one version of the Spacetime Interval (except you left off the squared term on "c") and it's actually the square of distance (just so we don't have to worry about taking the square root of a negative number). And it does incorporate both time and space but since they are actually different, we convert time to a distance that light travels in a given time.

 

[dextercioby]

The speed of light is a conversion factor in terms of units. You must pass somehow from seconds to meters and do that with something measured in meters/second. The second postulate of SR forces this conversion factor to be exactly c (and not c+/- 2%).

 

[dauto]

It is also possible to write the space-time interval as dτ² = dt² - 1/c²[dx² + dy² + dz²]

 

[nomadreid]

thank you, ghwellsjr and dextercioby. Yes, I have corrected the formula to having the c-squared and that we are looking at distance squared. That two sides of the equation should have the same units, and that the c² accomplishes this is clear, and that both space and time are implicitly taken into account in the calculation of ds² is also clear. However, this still bypasses my question: with this mechanism, we end up with ds² having the units of, for example, m². Yes, in a calculation this works, but spacetime is not a spatial distance and should hence not have the units of spatial distance, just as it would be odd to express a measurement of mass in units of electric charge: the two are fundamentally different. My own guess is either that
(a) there is some sort of implicit isomorphism at work here, so that we have a distance-squared
(dx²+dy²+dz²-c²*dt²) which relates to the measurement of a distance in a structure in space, and that this structure of space is isomorphic to a structure in spacetime, in which the units of the metric are no longer purely spatial.
Or,
(b) once the units have been all set to the same units, then these quantities are really unitless, merely computing comparative lengths of the space, the time, and the space-time vectors.

But I don't know whether either of these guesses are viable. I would be grateful for further enlightenment.

 

[nomadreid]

PS the reply of dauto just came in as I sent in my reply. Thanks, dauto; yes, using proper time (tau) has both sides of the equation expressed in time-squared units, but this then transforms my question from
(A) why spacetime-squared is in terms of spatial-distance-units squared, to
(B) why spacetime-squared is in terms of time-units squared.
Just as spacetime is different to space, it is also different to time. Again, both time and space are taken into account in the calculation of either mode of expression, but in neither does spacetime have its proper units distinct from space alone or time alone.
My dubious guesses (a) and (b) would also apply here.

 

[ghwellsjr]

If you start with your version of the spacetime interval and it evaluates to a positive number, then you can take the squareroot and you will have a value that is purely spacial. It's called a spacelike spacetime interval. If it's negative, then you can recalculate it using the form that dauto provided, take its squareroot and you will have a value that purely temporal. That one is called a timelike spacetime interval. If the value is non-zero, it's always either one or the other, never both. If it's zero, it's neither and it's called a null spacetime interval.

 

[nomadreid]

thanks, ghwellsjr. Interesting. I shall work further on this, starting from your explanation.

 

[phyti]

... However, this still bypasses my question: with this mechanism, we end up with ds² having the units of, for example, m². Yes, in a calculation this works, but spacetime is not a spatial distance and should hence not have the units of spatial distance, just as it would be odd to express a measurement of mass in units of electric charge: the two are fundamentally different. My own guess is either that
(a) there is some sort of implicit isomorphism at work here, so that we have a distance-squared
(dx²+dy²+dz²-c²*dt²) which relates to the measurement of a distance in a structure in space, and that this structure of space is isomorphic to a structure in spacetime, in which the units of the metric are no longer purely spatial.
Or,
(b) once the units have been all set to the same units, then these quantities are really unitless, merely computing comparative lengths of the space, the time, and the space-time vectors.

But I don't know whether either of these guesses are viable. I would be grateful for further enlightenment.

The evolution of the interval.

From the SR 1905 paper, the (equality) expression for the invariant interval between two events is

1. x1²+ x2²+ x3²= (ct)².

Nice and simple, with obvious meaning.
Minkowski, using complex notation, modifies the ct variable to ict. This allows a general form for the interval, that (mathematically) represents ct as a 4th dimension when expressed as

2. s² = x1² + x2² + x3² + x4².

More complex, allowing for misinterpretation, such as ‘moving in time’.
The s term can have ± values or = 0. Using a Minkowski diagram, the sign of the values only indicates a speed <, >, or = to c, i.e. time like, space like, or null (photon) world lines.

Spacetime is a measure of spatial units. If you examine the light clock, each tick is counting multiples of light motion within the clock. If you could unfold this distance into a straight line, it would represent ct. The Minkowski diagram is a plot of speed, vt/ct, or v/c, and is not a 2D plane.

 

[nomadreid]

Thanks, phyti. Your remark that it is ct, not t, that is a separate dimension is enlightening, as was your comment that spacetime is really a spatial measurement (implicitly: for a spacelike spacetime separation; mutatis mutandi for timelike spacetime measurements). Also enlightening was my attempt to give a counterexample; I kept finding myself guilty of assuming either simultaneity or motionlessness, both of which are unprovable. This helped not only my understanding but also my gut reaction against spacetime not having some units of its own.

 

[ghwellsjr]

From the SR 1905 paper, the (equality) expression for the invariant interval between two events is

1. x1²+ x2²+ x3²= (ct)².

Nice and simple, with obvious meaning.

Where does that occur in the SR 1905 paper? I couldn't find it or any discussion about the spacetime interval or about it being invariant.

I did find a similar equation in the middle of article 3 but it concerned the formula for the expansion of a spherical wave and had nothing to do with the invariant interval between two events. Einstein's use is nice and simple, with obvious meaning, but I don't see how it could be true for any two events, even if one of those events is at the origin.

 

[The_Duck]

PS the reply of dauto just came in as I sent in my reply. Thanks, dauto; yes, using proper time (tau) has both sides of the equation expressed in time-squared units, but this then transforms my question from
(A) why spacetime-squared is in terms of spatial-distance-units squared, to
(B) why spacetime-squared is in terms of time-units squared.
Just as spacetime is different to space, it is also different to time.

In special relativity, you should ideally measure distance and time in the same units, and you should also use these units for the spacetime interval. Note that "spacetime" is not a quantity, or a unit--what we are speaking of is called the spacetime *interval*. "Interval" here is supposed to mean "distance"--the spacetime interval defines a sort of distance between any two events, and it is perfectly fine to measure this distance in meters. The spacetime interval has the same units as distance and time for the same reason that all distances in the 2D Euclidean plane have the same units as distances along the x axis.

 

[nomadreid]

Thanks, The_Duck. A few comments, sentence by sentence:

In special relativity, you should ideally measure distance and time in the same units, and you should also use these units for the spacetime interval.

Yes, phyti's post also emphasizes this point.

Note that "spacetime" is not a quantity, or a unit...

Why isn't it a quantity? If it is measured or calculated, it is a quantity, no? In this case, although it is not a unit, it is measured in units, no?

The spacetime interval has the same units as distance and time for the same reason that all distances in the 2D Euclidean plane have the same units as distances along the x axis.

If you already have the two axes in distance units (as clarified by phyti), then fine. I was starting from the assumption that one of the axes in spacetime was time, in temporal units, and the other axis was space, in spatial units (as is usually the case in popular descriptions of spacetime), in which case the analogy to the 2D Euclidean plane would no longer work, since you have curves measured in the same units as those of the x-axis only if the units of the y-axis are the same as those of the x axis; however, if the axes are of different quantities measured in different units, this no longer works.

 

[Nugatory]

I was starting from the assumption that one of the axes in spacetime was time, in temporal units, and the other axis was space, in spatial units

As far as the mathematics of space-time are concerned, there is no distinction between the units of time-like intervals and space-like intervals.

We humans don't experience these intervals the same way, but that affects neither the mathematical structure nor the correspondence between that structure and what we do experience (in part because Minkowski mathematics does recognize a difference between time-like intervals and space-like intervals, it just doesn't need a different unit of measure to do so).

Measuring intervals on the t-axis in seconds and measuring intervals on the spatial axes in meters is like measuring distances on the y-axis in inches and distances on the x-axis in feet... It obscures without being any more correct.

 

[yuiop]

The evolution of the interval.

From the SR 1905 paper, the (equality) expression for the invariant interval between two events is

1. x1²+ x2²+ x3²= (ct)².

I agree with ghwellsjr. This expression is not the invariant interval between two events.
Using x2=x3=0 and rearranging yields:

$\frac{\Delta x1}{\Delta t} = c^2$

which is obviously only true for a light particle. If this is in the 1905 SR paper, then it was in the context of the speed of light being invariant.

 

[nomadreid]

Nugatory, thanks and a further question. If spatial and temporal units are considered essentially equivalent , would that make velocity (and hence the constant relating the two units) essentially unitless?

 

[phyti]

Where does that occur in the SR 1905 paper? I couldn't find it or any discussion about the spacetime interval or about it being invariant.

I did find a similar equation in the middle of article 3 but it concerned the formula for the expansion of a spherical wave and had nothing to do with the invariant interval between two events. Einstein's use is nice and simple, with obvious meaning, but I don't see how it could be true for any two events, even if one of those events is at the origin.

For this discussion, it should just be "interval" as in line 1. The expression you mentioned in par 3, is the one noted. The purpose was to compare (1) and (2) (post 9), showing the misinterpretation of (2) with "time" dressed up as a literal dimension. The classification of interval types, seems redundant, when they are so obvious in Minkowski diagrams. Mass moves at <c, light moves at c, and nothing known moves >c.
This idea of a time dimension just perpetuates the philosophical/scientific debates about the nature of time. More is learned about time by studying clocks and how they have been used. The conclusion is the same throughout history until now. Time is an alias for spatial motion, the Earth for millenia, and light for the last century.

From 'The Meaning of Relativity', Albert Einstein, 1956:
page 1.
"The experiences of an individual appear to us arranged in a series of events; in this series the single events which we remember appear to be ordered according to the criteria of "earlier" and "later", which cannot be analysed further. There exists, therefore, for the individual, an I-time, or subjective time."
page 31.
"The non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space coordinates with the time coordinate."
page 32.
"Finally, with Minkowski, we introduce in place of the real time co-ordinate l=ct, the imaginary time co-ordinate..."

 

[ghwellsjr]

Where does that occur in the SR 1905 paper? I couldn't find it or any discussion about the spacetime interval or about it being invariant.

I did find a similar equation in the middle of article 3 but it concerned the formula for the expansion of a spherical wave and had nothing to do with the invariant interval between two events. Einstein's use is nice and simple, with obvious meaning, but I don't see how it could be true for any two events, even if one of those events is at the origin.

For this discussion, it should just be "interval" as in line 1. The expression you mentioned in par 3, is the one noted. The purpose was to compare (1) and (2) (post 9), showing the misinterpretation of (2) with "time" dressed up as a literal dimension. The classification of interval types, seems redundant, when they are so obvious in Minkowski diagrams. Mass moves at <c, light moves at c, and nothing known moves >c.
This idea of a time dimension just perpetuates the philosophical/scientific debates about the nature of time. More is learned about time by studying clocks and how they have been used. The conclusion is the same throughout history until now. Time is an alias for spatial motion, the Earth for millenia, and light for the last century.

From 'The Meaning of Relativity', Albert Einstein, 1956:
page 1.
"The experiences of an individual appear to us arranged in a series of events; in this series the single events which we remember appear to be ordered according to the criteria of "earlier" and "later", which cannot be analysed further. There exists, therefore, for the individual, an I-time, or subjective time."
page 31.
"The non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space coordinates with the time coordinate."
page 32.
"Finally, with Minkowski, we introduce in place of the real time co-ordinate l=ct, the imaginary time co-ordinate..."

But couldn't you just as easily claim that the spatial dimensions are misinterpretations of the time coordinate as per dauto's post:

It is also possible to write the space-time interval as dτ² = dt² - 1/c²[dx² + dy² + dz²]

After all, the international standard for the meter is no longer based on a rigid rod or on a fraction of the size of the earth but rather on the distance that light travels during a specified interval of time. Apparently, the world's best scientists have become convinced that there is no such thing as a rigid rod. Why wouldn't you conclude based on that and since there has been so much debate concerning Length Contraction of so-called rigid rods that there is no legitimate way to construct a spatial coordinate system based on rigid rods but only on the measurement of distances using clocks to determine how far light has traveled during a specified time interval.

So why wouldn't you conclude your remarks with "Distance is an alias for light travel time and motion is just a fraction of the speed of light (or something similar).

 

[phyti]

But couldn't you just as easily claim that the spatial dimensions are misinterpretations of the time coordinate as per dauto's post:

Quote by dauto
It is also possible to write the space-time interval as dτ2 = dt2 - 1/c2[dx2 + dy2 + dz2]

You can rearrange x=ct, in different ways, depending on what variable is unknown, but that does not define them. It defines a relation among the variables.

After all, the international standard for the meter is no longer based on a rigid rod or on a fraction of the size of the Earth but rather on the distance that light travels during a specified interval of time.

If light travels a fixed distance, we can set a clock to run at different rates, and assign various “times” for the same distance. The distance is constant, the time is a convention, and thus variable.

Apparently, the world's best scientists have become convinced that there is no such thing as a rigid rod

They knew this long before 1900, when they discovered that heated matter expanded.

Why wouldn't you conclude based on that and since there has been so much debate concerning Length Contraction of so-called rigid rods that there is no legitimate way to construct a spatial coordinate system based on rigid rods but only on the measurement of distances using clocks to determine how far light has traveled during a specified time interval.

That was the point, that the “time” corresponded to a distance traveled by light.

Since lc and td scale length and time for the moving observer, by the same scale, 1/γ, the spatial coordinate system preserves relations between “time” and “space”, as if rigid rods exist, i.e. relative to that observer, the unit of time and length are constant.

So why wouldn't you conclude your remarks with "Distance is an alias for light travel time and motion is just a fraction of the speed of light (or something similar).

We already do that with light second, light year, etc., and expressing (speed) v as .6c for example.

The light clock counts units of light motion (t) that are simultaneous with the object motion. The numbers t1 & t2 are only events corresponding with the start and end of the motion, just as the marks on a ruler correspond to the ends of an object.

I can’t regard “time” as something fundamental which causes events. Observing and recording events using a clock (a standard event generator), and comparing to previous events is an operational definition of “time”, an accounting method. In its primitive form, a criminal in a prison marking a line on the wall for each day.


The Minkowski diagram shows explicitly, vertical ct axis (light motion) vs horizontal x-axis (object motion), v/c, an apples to apples comparison.

 

[ghwellsjr]

I can’t regard “time” as something fundamental which causes events.

Do you regard "distance" as something fundamental which causes events?

 

[phyti]

Do you regard "distance" as something fundamental which causes events?

No,and "distance" is not used in that sense, but "time" is. Consider the expression, v=h-gt, (the speed v of an object, released from a height h, is a function of time t. If you took pics of the object for a sequence of clock ticks, the ticks would correspond to varying heights of the object. Thus in reality you are comparing heights that vary exponentially with ticks that vary linearly, and conclude the parabolic path is gravity accelerating the object toward the Earth center.
We know it's gravity, not time, that accelerates the object.
Just as Einstein states in the 1905 paper, the time of an event is the simultaneous state (tick) of the clock, but notice the time is always after the fact/event/awareness, therefore it can't be a cause.

 

[ghwellsjr]

No,and "distance" is not used in that sense, but "time" is. Consider the expression, v=h-gt, (the speed v of an object, released from a height h, is a function of time t. If you took pics of the object for a sequence of clock ticks, the ticks would correspond to varying heights of the object. Thus in reality you are comparing heights that vary exponentially with ticks that vary linearly, and conclude the parabolic path is gravity accelerating the object toward the Earth center.
We know it's gravity, not time, that accelerates the object.
Just as Einstein states in the 1905 paper, the time of an event is the simultaneous state (tick) of the clock, but notice the time is always after the fact/event/awareness, therefore it can't be a cause.

I'm not sure anyone ever considered time to be a cause.

Let me rephrase the question: Do you regard "distance" as being fundamental in some sense that "time" is not?

 

[phyti]

I'm not sure anyone ever considered time to be a cause.

Let me rephrase the question: Do you regard "distance" as being fundamental in some sense that "time" is not?

Space although intangible, makes its presence known. An object is here, another there. It requires energy to move from one location to another. Physics has to allow for it.
Time as shown in many examples is a human contrivance/tool/methodology, to put events in order, and provide a measurable interval, for whatever purpose selected. Historically, it can be any periodic event, some form of clock, the moon, rotation of the earth, atomic vibration, etc, all involving some form of motion.
We could substitute "SR was published 108 yrs ago", with "SR was published 108 Earth orbits ago".
Distance is labelled as "time".

If you observed that all motion of material objects beyond yourself stopped, you would still be sensing new photons, therefore you would consider "time" to be passing. If the photons (motions) ceased, there would be no objects, no memory of a previous state, i.e. the equivalent of death.
Don't you think this reduces "time" to motion (a changing spatial position)?

 

[nomadreid]

If I may insert a question that is a real question, not a rhetorical one (even if the question may appear silly): if time is reducible to distance, how is the "arrow of time" translated? That is, on the macro scale, time has the property of going only in one direction, while change of spatial position need not. Is the "arrow of time" translated then as a sort of predestination for the macro position changes according to the forces in the universe? Thanks for any insight.

 

[phyti]

If I may insert a question that is a real question, not a rhetorical one (even if the question may appear silly): if time is reducible to distance, how is the "arrow of time" translated? That is, on the macro scale, time has the property of going only in one direction, while change of spatial position need not. Is the "arrow of time" translated then as a sort of predestination for the macro position changes according to the forces in the universe? Thanks for any insight.

Spatial dimensions have direction (vectors). Time is only a number (magnitude/scalar). It is cumulative, thus always increasing. All events are historical (assigned times after awareness), so the next event is associated with the next available “time” or clock tick. You can liken this to an “arrow”, but it has no more significance than the imagined lines connecting the stars to form astrological signs. If you assign sequential numbers to customers in line at a deli, what significance do the numbers have, beyond an orderly service?

 

[ghwellsjr]

Don't you think this reduces "time" to motion (a changing spatial position)?

Motion is a changing spatial position but I don't understand why you have picked out "time" as the parameter to be reduced as opposed to "spatial position" to be reduced. Motion has both parameters, "spatial position" divided by "time" or "spatial position" as a function of "time". I'm totally confused about what you are saying and I don't know what it has to do with relativity.

 

[phyti]

Motion is a changing spatial position but I don't understand why you have picked out "time" as the parameter to be reduced as opposed to "spatial position" to be reduced. Motion has both parameters, "spatial position" divided by "time" or "spatial position" as a function of "time". I'm totally confused about what you are saying and I don't know what it has to do with relativity.

It's about "time", wherever it's used.
The question is: "What is the light clock counting with each tick?".
It is counting a multiple of the vertical light path to and from the mirror, i.e. a motion or distance.

 

[ghwellsjr]

It's about "time", wherever it's used.
The question is: "What is the light clock counting with each tick?".
It is counting a multiple of the vertical light path to and from the mirror, i.e. a motion or distance.

Some clocks (like light clocks) do involve motion and distance but not all clocks. The international standard for the second does not involve any motion but rather is defined in terms of specific energy levels of a cesium 133 atom. We could also make a clock based on radioactive decay which does not involve any motion.

So I don't know why you continue to equate or link (or whatever) time with motion or distance. Furthermore, even if you want to use a light clock as your example, I don't understand why you insist that time is fundamentally or really or basically (or whatever) motion or distance rather than saying that distance is fundamentally motion or time.

If we have previously defined the speed of light to be some particular value, then we could define our unit of time in terms of how long it takes light to travel a previously defined unit of distance or we could define our unit of distance in terms of how far light travels in a previously defined unit of time, couldn't we? Or we could define our unit of time independently of the speed of light (like we do now) and we could define our unit of distance independently of the speed of light (like we used to do between 1960 and 1983 when it was based on the wavelength of krypton-86) and then we can measure the speed of light. In your mind, what makes one of these more basic or fundamental or logical or important or significant (or whatever) than the other?

 

[phyti]

Some clocks (like light clocks) do involve motion and distance but not all clocks. The international standard for the second does not involve any motion but rather is defined in terms of specific energy levels of a cesium 133 atom. We could also make a clock based on radioactive decay which does not involve any motion.

So I don't know why you continue to equate or link (or whatever) time with motion or distance. Furthermore, even if you want to use a light clock as your example, I don't understand why you insist that time is fundamentally or really or basically (or whatever) motion or distance rather than saying that distance is fundamentally motion or time.

If we have previously defined the speed of light to be some particular value, then we could define our unit of time in terms of how long it takes light to travel a previously defined unit of distance or we could define our unit of distance in terms of how far light travels in a previously defined unit of time, couldn't we? Or we could define our unit of time independently of the speed of light (like we do now) and we could define our unit of distance independently of the speed of light (like we used to do between 1960 and 1983 when it was based on the wavelength of krypton-86) and then we can measure the speed of light. In your mind, what makes one of these more basic or fundamental or logical or important or significant (or whatever) than the other?

We are just rehashing the same stuff. Wikipedia has a good history of the “second”.

The red text looks like circular reasoning.

All clocks are based on motion. Historically the motion of matter in a device, which is a subdivision of a day or year. The energy level transitions are only observable by emission. It is still a definition of a light second, i.e. a distance traveled by light. The light clock had to wait for the discovery that light speed is not instantaneous.
All definitions of time are derived relations of distance/speed. Distance is a measure of physical space between positions. Speed is a rate of change of position. It’s all about position. There is no known fundamental entity labeled “time”. This is in agreement with Einstein stating time is subjective and dependent on observer speed.

 

[ghwellsjr]

We are just rehashing the same stuff.

I think you have provided more clarification in this post than others, thanks.

Wikipedia has a good history of the “second”.

Yes, they do and here is what they say about the current definition of the second:

Under the International System of Units (via the International Committee for Weights and Measures, or CIPM), since 1967 the second has been defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

I don't see any statement regarding distance, motion or the speed of light in this definition. Where do you see it?

The red text looks like circular reasoning.

There are three parameters involved:
1) The speed of light.
2) The unit of distance.
3) The unit of time.

We can define any two independently and then define or measure the remaining one in terms of the other two. I'm just asking you why you feel that the only valid way is to define the unit of distance and the speed of light independently and then define the unit of time in terms of the other two, as opposed to either of the other two options. (And why you don't seem to understand that the current definitions define the speed of light and the unit of time independently and then define the unit of distance based on the other two.)

All clocks are based on motion. Historically the motion of matter in a device, which is a subdivision of a day or year.

Historically, if you go back far enough, yes, but not now.

The energy level transitions are only observable by emission.

True, but so what?

It is still a definition of a light second, i.e. a distance traveled by light.

Really? What distance? What is the value of the light second that you believe is involved? I have no idea what you are referring to.

The light clock had to wait for the discovery that light speed is not instantaneous.

I agreed that the light clock uses a distance and motion in its definition of time but now we're talking about clocks that don't use distance or motion so let's don't keep going around this circle.

All definitions of time are derived relations of distance/speed.

If you're going to make this claim, you need to explain what you mean with regard to our current definition of the second. Please provide the details because I don't understand where distance or speed is involved.

Distance is a measure of physical space between positions. Speed is a rate of change of position. It’s all about position. There is no known fundamental entity labeled “time”. This is in agreement with Einstein stating time is subjective and dependent on observer speed.

Where did Einstein make that statement? Please provide an online reference and point to the specific location and/or provide a direct quote.