Learn About Spacetime

In this episode of the I Can't Sleep Podcast, fall asleep learning about spacetime. For a moment there I was optimistic that I was going to understand this concept after a good reading of it. Yeah, I was wrong about that because I think I confused myself even more with all the math it tries to explain. Well, happy sleeping!

Transcript

Welcome to the I Can't Sleep Podcast,

Where I read random articles from across the web to bore you to sleep with my soothing voice.

I'm your host,

Benjamin Boster.

Today's episode is from a Wikipedia article titled,

Spacetime.

In physics,

Spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

Spacetime diagrams are useful in visualizing and understanding relativistic effects,

Such as how different observers perceive where and when events occur.

Until the turn of the 20th century,

The assumption had been that the three-dimensional geometry of the universe,

Its description in terms of locations,

Shapes,

Distances,

And directions,

Was distinct from time,

The measurement of when events occur within the universe.

However,

Space and time took on new meanings with the Lorentz transformation and special theory of relativity.

In 1908,

Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum,

Now known as Minkowski space.

This interpretation proved vital to the general theory of relativity,

Wherein spacetime is curved by mass and energy.

Non-relativistic classical mechanics treats time as a universal quantity of measurement,

Which is uniform throughout space and separate from space.

Classical mechanics assume that time has a constant rate of passage,

Independent of the observer's state of motion or anything external.

Furthermore,

It assumes that space is Euclidean.

It assumes that space follows the geometry of common sense.

In the context of special relativity,

Time cannot be separated from the three dimensions of space,

Because the observed rate at which time passes for an object depends on the object's velocity relative to the observer.

General relativity also provides an explanation of how gravitational fields can slow the passage of time for an object,

As seen by an observer outside the field.

In ordinary space,

A position is specified by three numbers,

Known as dimensions.

In the Cartesian coordinate system,

These are called x,

Y,

And z.

A position in spacetime is called an event and requires four numbers to be specified,

The three-dimensional location in space plus the position in time.

An event is represented by a set of coordinates x,

Y,

Z,

And t.

Spacetime is thus four-dimensional.

Unlike the analogies used in popular writings to explain events,

Such as firecrackers or sparks,

Mathematical events have zero duration and represent a single point in spacetime.

Although it is possible to be in motion relative to the popping of a firecracker or a spark,

It is not possible for an observer to be in motion relative to an event.

The path of a particle through spacetime can be considered to be a succession of events.

The series of events can be linked together to form a line,

Which represents a particle's progress through spacetime.

That line is called the particle's world line.

Mathematically,

Spacetime is a manifold,

Which is to say it appears locally flat near each point in the same way that,

At small enough scales,

The surface of a globe appears flat.

A scale factor conventionally called the speed of light relates distances measured in space with distances measured in time.

The magnitude of this scale factor,

Nearly 300,

000 kilometers or 190,

000 miles in space,

Being equivalent to one second in time,

Along with the fact that spacetime is a manifold,

Implies that at ordinary non-relativistic speeds and at ordinary human scale distances,

There's little that humans might observe which is noticeably different from what they might observe if the world were Euclidean.

It was only with the advent of sensitive scientific measurements in the mid-1800s,

Such as the Fizeau experiment and the Michelin-Morley experiment,

That puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space.

In special relativity,

An observer will,

In most cases,

Mean a frame of reference from which a set of objects or events is being measured.

This usage differs significantly from the ordinary English meaning of the term.

Reference frames are inherently non-local constructs,

And according to this usage of the term,

It does not make sense to speak of an observer as having a location.

For example,

Imagine that a frame is equipped with a dense lattice of clocks,

Synchronized within the reference frame,

That extends indefinitely throughout the three dimensions of space.

Any specific location within the lattice is not important.

The latticework of clocks is used to determine the time and position of events taking place within the whole frame.

The term observer refers to the entire ensemble of clocks associated with one inertial frame of reference.

In this idealized case,

Every point in space has a clock associated with it,

And thus the clocks register each event instantly,

With no time delay between an event and its recording.

A real observer,

However,

Will see a delay between the emission of a signal and its detection due to the speed of light.

To synchronize the clocks and data reduction following an experiment,

The time when a signal is received will be corrected to reflect its actual time,

Were it to have been recorded by an idealized lattice of clocks.

In many books on special relativity,

Especially older ones,

The word observer is used in the more ordinary sense of the word.

It is usually clear from context which meaning has been adopted.

Physicists distinguish between what one measures or observes,

After one has factored out signal propagation delays,

Versus what one visually sees without such corrections.

Physicists such corrections.

Failure to understand the difference between what one measures or observes,

Versus what one sees,

Is the source of much error among beginning students of relativity.

By the mid-1800s,

Various experiments,

Such as the observation of the Orego spot,

And differential measurements of the speed of light and air versus water,

Were considered to have proven the wave nature of light,

As opposed to a corpuscular theory.

Propagation of waves was then assumed to require the existence of a waving medium.

In the case of light waves,

This was considered to be a hypothetical luminiferous aether.

However,

The various attempts to establish the properties of this hypothetical medium yielded contradictory results.

For example,

The Fizeau experiment of 1851,

Conducted by French physicist Hippolyte Fizeau,

Demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of water by an amount dependent on the water's index of refraction.

Among other issues,

The dependence of the partial aether dragging implied by this experiment on the index of refraction,

Which is dependent on wavelength,

Led to the unpalatable conclusion that aether simultaneously flows at different speeds for different colors of light.

The famous Michelson-Morley experiment of 1887 showed no differential influence of Earth's motions through the hypothetical aether on the speed of light,

And the most likely explanation,

Complete aether dragging,

Was in conflict with the observation of stellar aberration.

George Francis Fitzgerald in 1889 and Hendrick Lawrence in 1892 independently proposed that material bodies traveling through the fixed aether were physically affected by their passage,

Contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson-Morley experiment.

No length changes occur in directions transverse to the direction of motion.

By 1904,

Lawrence had expanded his theory such that he had arrived at equations formally identical with those that Einstein was to derive later,

I.

E.

The Lawrence transformation.

As a theory of dynamics,

The study of forces and torques and their effect on motion,

His theory assumed actual physical deformations of the physical constituents of matter.

Lawrence's equations predicted a quantity that he called local time with which he could explain the aberration of light,

The Fizeau experiment,

And other phenomena.

Henri Poincaré was the first to combine space and time into spacetime.

He argued in 1898 that the simultaneity of two events is a matter of convention.

In 1900,

He recognized that Lawrence's local time is actually what is indicated by moving clocks by applying an explicitly operational definition of clock synchronization,

Assuming constant light speed.

In 1900 and 1904,

He suggested that inherent undetectability of the ether by emphasizing the validity of what he called the principle of relativity,

And in 1905-1906,

He mathematically perfected Lawrence's theory of electrons in order to bring it into accordance with the postulate of relativity.

While discussing various hypotheses on Lawrence's invariant gravitation,

He introduced the innovative concept of a four-dimensional spacetime by defining various four vectors,

Namely four-position,

Four-velocity,

And four-force.

He did not pursue the four-dimensional formalism in subsequent papers,

However,

Stating that this line of research seemed to entail great pain for limited profit,

Ultimately concluding that three-dimensional language seems the best suited to the description of our world.

Furthermore,

Even as late as 1909,

Poincaré continued to describe the dynamical interpretation of the Lawrence transform.

In 1905,

Albert Einstein analyzed special relativity in terms of kinematics,

The study of moving bodies without reference to forces,

Rather than dynamics.

His results were mathematically equivalent to those of Lawrence and Poincaré.

He obtained them by recognizing that the entire theory can be built upon two postulates,

The principle of relativity and the principle of the consistency of light speed.

His work was filled with vivid imagery involving the exchange of light signals between clocks and motion,

Careful measurements of the lengths of moving rods,

And other such examples.

In addition,

Einstein in 1905 superseded previous attempts of an electromagnetic mass-energy relation by introducing the general equivalence of mass and energy,

Which was instrumental for his subsequent formulation of the equivalence principle in 1907,

Which declares the equivalence of inertial and gravitational mass.

By using the mass-energy equivalence,

Einstein showed,

In addition,

That the gravitational mass of a body is proportional to its energy content,

Which was one of the early results in developing general relativity.

While it would appear that he did not at first think geometrically about spacetime,

In the further development of general relativity,

Einstein fully incorporated the spacetime formalism.

When Einstein published in 1905,

Another of his competitors,

His former mathematics professor Hermann Minkowski,

Had also arrived at most of the basic elements of special relativity.

Max Born recounted a meeting he had made with Minkowski,

Seeking to be Minkowski's student collaborator.

I went to Cologne,

Met Minkowski,

And heard his celebrated lecture,

Space and Time,

Delivered on the 2nd of September 1908.

He told me later that it came to him as a great shock when Einstein published his paper,

In which the equivalence of the different local times of observers moving relative to each other was pronounced.

For he had reached the same conclusions independently,

But did not publish them,

Because he wished first to work out the mathematical structure in all its splendor.

He never made a priority claim and always gave Einstein his full share in the great discovery.

Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments,

At least since the summer of 1905,

When Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz,

Poincaré,

Et al.

Minkowski saw Einstein's work as an extension of Lorentz's and was most directly influenced by Poincaré.

On the 5th of November 1907,

A little more than a year before his death,

Minkowski introduced his geometric interpretation of space-time in a lecture to the Göttingen Mathematical Society with the title,

The Relativity Principle.

On the 21st of September 1908,

Minkowski presented his famous talk,

Space and Time,

To the German Society of Scientists and Physicists.

The opening words of Space and Time include Minkowski's famous statement that,

Henceforth,

Space for itself and time for itself shall completely reduce to a mere shadow,

And only some sort of union of the two shall preserve independence.

Space and Time included the first public presentation of space-time diagrams,

And included a remarkable demonstration that the concept of the invariant interval,

Along with the empirical observation that the speed of light is finite,

Allows derivation of the entirety of special relativity.

The space-time concept and the Lorentz group are closely connected to certain types of sphere,

Hyperbolic,

Or conformal geometries,

And their transformation groups already developed in the 19th century,

In which invariant intervals analogous to the space-time interval are used.

Einstein,

For his part,

Was initially dismissive of Minkowski's geometric interpretation of special relativity,

Regarding it as superfluous learnedness.

However,

In order to complete his search for general relativity that started in 1907,

The geometric interpretation of relativity proved to be vital,

And in 1916 Einstein fully acknowledged his indebtedness to Minkowski,

Whose interpretation greatly facilitated the transition to general relativity.

Since there are other types of space-time,

Such as the curved space-time of general relativity,

The space-time of special relativity is today known as Minkowski space-time.

In three dimensions,

The distance between two points can be defined using the Pythagorean theorem.

Although two viewers may measure the x,

Y,

And z position of the two points using different coordinate systems,

The distance between the points will be the same for both,

Assuming that they are measuring using the same units.

The distance is invariant.

In special relativity,

However,

The distance between two points is no longer the same if measured by two different observers when one of the observers is moving,

Because of Lorentz contraction.

The situation is even more complicated if the two points are separated in time as well as in space.

For example,

If one observer sees two events occur at the same place but at different times,

A person moving with respect to the first observer will see the two events occurring at different places,

Because from their point of view they are stationary and the position of the event is receding or approaching.

Thus,

A different measure must be used to measure the effective distance between two events.

In four-dimensional space-time,

The analog to distance is the interval.

Although time comes in as a fourth dimension,

It is treated differently than the spatial dimensions.

Minkowski space hence differs in important respects from four-dimensional Euclidean space.

The fundamental reason for merging space and time into space-time is that space and time are separately not invariant,

Which is to say that under the proper conditions,

Different observers will disagree on the length of time between two events,

Because of time dilation,

Or the distance between the two events,

Because of length contraction.

But special relativity provides a new invariant called the space-time interval,

Which combines distances in space and in time.

All observers who measure the time and distance between any two events will end up computing the same space-time interval.

Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts.

If an observer in frame s measures a clock at rest in frame s-prime as running slower than his,

While s-prime is moving at speed v in s,

Then the principle of relativity requires that an observer in frame s-prime likewise measures a clock in frame s.

Moving at speed minus v in s-prime as running slower than hers.

How two clocks can run both slower than the other is an important question that goes to the heart of understanding special relativity.

This apparent contradiction stems from not correctly taking into account the different settings of the necessary related measurements.

These settings allow for a consistent explanation of the only apparent contradiction.

It is not about the abstract ticking of two identical clocks,

But about how to measure in one frame the temporal distance of two ticks of a moving clock.

It turns out that in mutually observing the duration between ticks of clocks,

Each moving in the respective frame,

Different sets of clocks must be involved.

In order to measure in frame s the tick duration of a moving clock w-prime at rest in s-prime,

One uses two additional synchronized clocks w1 and w2 at rest and two arbitrarily fixed points in s with the spatial distance d.

Two events can be defined by the condition two clocks are simultaneously at one place,

I.

E.

When w-prime passes each w1 and w2.

For both events,

The two readings of the co-located clocks are recorded.

The difference of the two readings of w1 and w2 is the temporal distance of the two events in s and their spatial distance is d.

The difference of the two readings of w-prime is the temporal distance of the two events in s-prime.

In s-prime these events are only separated in time.

They happen at the same place in s-prime.

Because of the invariance of the spacetime intervals spanned by these two events and the non-zero spatial separation d in s,

The temporal distance in s-prime must be smaller than the one in s.

The smaller temporal distance between the two events resulting from the readings of the moving clock w-prime belongs to the slower running clock w-prime.

Conversely,

For judging and frame s-prime,

The temporal distance of two events on a moving clock w at rest in s,

One needs two clocks at rest in s-prime.

In this comparison,

The clock w is moving by with velocity negative v.

Recording again the four readings for the events,

Defined by two clocks simultaneously at one place,

Results in the analogous temporal distances of the two events,

Now temporally and spatially separated in s-prime and only temporarily separated but co-located in s.

To keep the spacetime interval invariant,

The temporal distance in s must be smaller than in s-prime because of the spatial separation of the events in s-prime.

Now clock w is observed to run slower.

The necessary recordings for the two judgments with one moving clock and two clocks at rest and respectively s or s-prime involves two different sets,

Each with three clocks.

Since there are different sets of clocks involved in the measurements,

There is no inherent necessity that the measurements be reciprocally consistent,

Such that if one observer measures the moving clock to be slow,

The other observer measures the one's clock to be fast.

There's a figure in the article that illustrates the previous discussion of mutual time dilation with Minkowski's diagrams.

The upper picture reflects the measurements as seen from frame s at rest with unprimed rectangular axes and frame s-prime moving with v greater than zero,

Coordinated by primed oblique axes slanted to the right.

The lower picture shows frame s-prime at rest with primed rectangular coordinates and frame s moving with negative v less than zero with unprimed oblique axes slanted to the left.

Each line drawn parallel to a spatial axis x x-prime represents a line of simultaneity.

All events on such a line have the same time value c t c t-prime.

Likewise,

Each line drawn parallel to a temporal axis c t c t-prime represents a line of equal spatial coordinates values x x-prime.

One may designate in both pictures the origin o equals o-prime as the event where the respective moving clock is co-located with the first clock at rest in both comparisons.

Obviously for this event the reasons on both clocks and both comparisons are zero.

As a consequence the worldliness of the moving clocks are the slanted to the right c t-prime axis upper pictures clock w-prime and the slanted to the left c t-axis lower pictures clock w.

The worldliness of w-one and w-prime one are the corresponding vertical time axes c t in the upper pictures and c t-prime in the lower pictures.

In the upper picture the place for w-two is taken to be a x greater than zero and thus the world line not shown in the pictures of this clock intersects the world line of the moving clock the c t-prime axis in the event labeled a where two clocks are simultaneously at one place.

In the lower picture the place for w-prime two is taken to be c x-prime less than zero and so in this measurement the moving clock w passes w-prime two in the event c.

In the upper picture the c t-coordinate a t of the event a the reading of w-two is labeled b thus giving the elapsed time between the two events measured with w-one and w-two prime as o b.

For a comparison the length of the time interval o a measured with w-prime must be transferred to the scale of the c t-axis.

This is done by the invariant hyperbola through a connecting all events with the same space-time interval from the origin as a.

This yields the event c on the c t-axis and obviously o c is less than o b the moving clock w-prime runs slower.

To show the mutual time dilation immediately in the upper picture the event d may be constructed as the event at x-prime equals zero the location of clock w-prime and s-prime that is simultaneous to c o c has equal space-time intervals as o a and s-prime.

This shows that the time interval o d is longer than o a showing that the moving clock runs slower.

In the lower picture the frame s is moving with velocity minus v in the frame s-prime at rest.

The world line of a clock w is the c t-axis slanted to the left.

The world line of w-prime one is a vertical c t-prime axis and the world line of w-prime two is a vertical through event c with c t-prime coordinate d.

The invariant hyperbola through event c scales to the time interval o c to o a which is shorter than o d.

Also b is constructed similar to d in the upper pictures as simultaneous to a in s at x equals zero.

The result o b greater than o c corresponds again to above.

The word measure is important.

In classical physics an observer cannot affect an observed object but the object's state of motion can affect the observer's observations of the object.

Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of paradoxes.

These paradoxes are in fact ill-posed problems resulting from our unfamiliarity with velocities comparable to the speed of light.

The remedy is to solve many problems in special relativity and to become familiar with its so-called counterintuitive predictions.

The geometrical approach to studying space-time is considered one of the best methods for developing a modern intuition.

The twin paradox is a thought experiment involving identical twins one of whom makes a journey into space in a high-speed rocket returning home to find that the twin who remained on earth has aged more.

This result appears puzzling because each twin observes the other twin as moving and so at first glance it would appear that each should find the other to have aged less.

The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock.

Nevertheless the twin paradox is not a true paradox because it is easily understood within the context of special relativity.

The impression that a paradox exists stems from a misunderstanding of what special relativity states.

Special relativity does not declare all frames of reference to be equivalent,

Only inertial frames.

The traveling twin's frame is not inertial during periods when she is accelerating.

Furthermore the difference between the twins is observationally detectable.

The traveling twin needs to fire her rockets to be able to return home while the stay-at-home twin does not.

These distinctions should result in a difference in the twins ages.

The space-time diagram of the figure above represents the simple case of a twin going straight out along the x-axis and immediately turning back.

From the standpoint of the stay-at-home twin there is nothing puzzling about the twin paradox at all.

The proper time measured along the traveling twin's world line from O to C plus the proper time measured from C to B is less than the stay-at-home twin's proper time measured from O to A to B.

More complex trajectories require integrating the proper time between the respective events along the curve,

I.

E.

The path integral,

To calculate the total amount of proper time experienced by the traveling twin.

Complications arise if the twin paradox is analyzed from the traveling twin's point of view.

Weiss's nomenclature designating the stay-at-home twin as Terence and the traveling twin as Stella is hereafter used.

Stella is not in an inertial frame.

Given this fact it is sometimes incorrectly stated that full resolution of the twin paradox requires general relativity.

A pure SR analysis would be as follows.

Analyzed in Stella's rest frame she is motionless for the entire trip.

When she fires her rockets for the turnaround she experiences a pseudo force which resembles a gravitational force.

Figures 2.

6 and 2.

11 illustrate the concept of lines,

Planes,

Of simultaneity.

Lines parallel to the observer's x-axis,

X-y-plane,

Represent sets of events that are simultaneous in the observer frame.

In figure 2.

11 the blue lines connect events on Terence's world line which,

From Stella's point of view,

Are simultaneous with events on her world line.

Terence,

In turn,

Would observe a set of horizontal lines of simultaneity.

Throughout both the outbound and the inbound legs of Stella's journey she measures Terence's clocks as running slower than her own.

But during the turnaround,

I.

E.

Between the bold blue lines in the figure,

A shift takes place in the angle of her lines of simultaneity,

Corresponding to a rapid skip over of the events in Terence's world line that Stella considers to be simultaneous with her own.

Therefore,

At the end of her trip,

Stella finds that Terence has aged more than she has.

Although general relativity is not required to analyze the twin paradox,

Application of the equivalence principle of general relativity does provide some additional insight into the subject.

Stella is not stationary in an inertial frame.

Analyzed in Stella's rest frame,

She is motionless for the entire trip.

When she is coasting her rest frame is inertial,

And Terence's clock will appear to run slow.

But when she fires her rockets for the turnaround,

Her rest frame is an accelerated frame,

And she experiences a force which is pushing her as if she were in a gravitational field.

Terence will appear to be high up in that field,

And because of gravitational time dilation,

His clock will appear to run fast.

So much so that the net result will be that Terence has aged more than Stella when they are back together.

The theoretical arguments predicting gravitational time dilation are not exclusive to general relativity.

Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence,

Including Newton's theory.

Reviews

Can't think of anything more boring and sleep inducing than adding math 😴 to the equation 😳Good job Benjamin👏

It did its job; I was asleep faster than spacetime!! Thanks!!