Understanding Space-time

In the theory of relativity, time plays a key role. In order to properly understand how the theory of relativity says time should behave, we must come up with a means of mentally visualizing time in the way that the theory of relativity treats it. This involves becoming aware of a concept that the theory of relativity refers to as space-time. As the means of doing so we are to picture in our minds an inflating balloon. This inflating balloon represents a 2-dimensional universe undergoing the passage of time. Evenly distributed across the surface of the balloon and moving along with it is an assortment of dots placed beforehand by means of a marker. These dots represent bodies within the 2-dimensional universe, undergoing the passage of time along with it. Observe, if you will, that as the balloon inflates, the dots in response to that inflation move outward in straight lines from the centerpoint of the balloon. Of what importance are the assumed paths of these dots to us?

Time is linear. In this sense time has a 'direction': it moves away from the past and toward the future. When we associate the inflation of the balloon with the passage of time, we are assigning a physical direction to time: we understand toward the centerpoint of the balloon as being into the past, and away from the centerpoint of the balloon as being into the future. The lines formed by the paths of the dots, when brought inward, converge at the centerpoint of the balloon. How would a 2-dimensional creature within the 2-dimensional universe represented by the surface of the balloon experience these lines? From within the surface of the balloon, take note, these lines simply do not exist! The directions that the dots move in as the balloon inflates are equally nonexistent, for the direction any given dot will be found to move outward into is a direction perpendicular to the surface of the balloon! To a 2-dimensional creature within the surface of the balloon, 'past' and 'future' are not spatial directions as they are in terms of how we perceive them: given that time is motion, the 2-dimensional creature experiences the physical motion of the balloon (motion in a direction the creature cannot detect) as the 'motion' associated with the passage of time. It is this union of space and time - space-time - by which we think of time as being a spatial dimension, and in doing so make possible an approach that allows us to view the universe from as basic a level as possible.

How does time operate within a space-time universe? The space-time approach to time, take note, allows one to relate to time in ways not previously possible. As the means of learning to take advantage of this opportunity, imagine the balloon before us to shrink to the size of a dimensionless point. It is here that all time begins. We will now assume that inflation of the balloon begins, marking the beginning of time for the 2-dimensional universe that the surface of the balloon represents - an event we will refer to as the point-out. After the occurrence of the point-out, time proceeds to advance at a constant rate. As time passes, the balloon increases in size. In fact, there lies a direct relationship between the balloon's size and the amount of time to pass since the point-out. The relationship can be successfully described by stating that the greater the amount of time to pass since the point-out, the larger in size the balloon will be observed to be. Time consists of a past, a present, and a future. The passage of time, furthermore, can be described as occurring by means of a passing present moment. As a rule, there can be no more than a single passing present moment. The notion of more than one present moment being possible has no meaning here. In the inflating balloon model we are working with, the surface of the balloon represents that single passing present moment.

Take note, then, that because we are treating time like a spatial direction, the concept of a passing present moment takes on extra significance. To understand this extra significance, consider the notion that for any particular given present moment in time (in terms of the inflation of the balloon), a balloon size exists that corresponds to that moment in time. Of what importance is balloon size? To speak of a balloon's size is another way of referring to the physical location of that balloon's surface. What does this tell us? It tells us that for every instant in time beginning at and following the occurrence of the point-out, there is a specific physical location of the balloon that corresponds to that instant in time. This arrangement is similar to space-time in many ways, which we are to make note of. The reasoning behind the concept of space-time is as follows: given that each instant in time of the balloon's inflation corresponds to a physical location of the surface of the balloon, why not refer to instants in time by means of those physical locations? In the realm of space-time, you see, time as we know it does not exist: everything is spatial. All moments in time, being of spatial substance, exist simultaneously. To think in terms of space-time means to experience time independently of a passing present moment - which is why the idea of a passing present moment is not applicable to space-time. Space-time is, in a word, "timeless". How, then, can we mentally visualize the space-time entity that we are discussing? We are to imagine all instants in time of the balloon's expansion to be simultaneously given physical substance. The result: a solid sphere! This makes perfect sense once you realize that the balloon starts out as a dimensionless point, and then continues to 'add on' increasingly larger hollow layers, that become part of the current accumulation. What should we refer to this formation as? This formation is to be referred to by the term that describes it: 3-dimensional space-time. By definition, 3-dimensional space-time refers to a space-time continuum that requires 3 physical dimensions of space to be constructed. The structure of 3-dimensional space-time, also, is, as we've learned, a solid sphere. Why are 3 dimensions required? Two of the physical dimensions are attributed to the 2-dimensional universe (the surface of the balloon). The remaining physical dimension is attributed to the 2-dimensional universe's time (as physically manifested).

We are as of now to imagine the surface of the balloon we have been working with to be marked by a 2-dimensional grid system of lines, much like the lines you will find on any globe. It is by means of this 2-dimensional coordinate system that we can refer to any location within the 2-dimensional universe. No location within the 2-dimensional universe exists, take note, that can't be designated by means of 2 coordinates. Let us take this a step further. The key to understanding the "timelessness" we have attributed to space-time is familiarity with the use and application of coordinates. Our means of working with coordinates will be 3-dimensional space-time. By definition, 3-dimensional space-time is a 3-dimensional coordinate system: any location within it can be designated by means of 3 coordinates. The first 2 coordinates designate a physical location within the 2-dimensional universe (where on the surface of the balloon to go). The final third coordinate, in turn, designates the instant in time associated with the location (the size of the balloon). Designating 3 such coordinates is the equivalent of what is called a space-time event. The clear advantage to the "timelessness" we have attributed to space-time is that events in time can be plotted in the same manner one would plot a point on a graph. When viewing things in this manner, one need not even think of the concept of time: when time is a spatial dimension, finding an event in time is only a matter of designating coordinates. It is being able to treat time in this way that makes all the difference.

According to the theory of relativity, mass curves space-time. As the first step in understanding this topic, we will now discuss its most basic aspects. Mass refers to matter. The specific type of matter being referred to in the above statement would in this situation be the mass of a celestial body such as a planet or a star. Next, we are informed that mass 'curves' space-time. As the means of understanding this we are to first draw attention to our use of an inflating balloon to represent a 2-dimensional universe. The decision to use an inflating balloon for this purpose was no accident. A balloon, take note, is an expanding elastic surface. It so happens that according to the approach that the theory of relativity exercises, an expanding elastic surface is in many ways an accurate description of what a universe is. To picture how mass 'curves' a surface, think of an outstretched thin latex sheet, held at its edges. Next imagine placing a baseball on the sheet. The baseball's weight, it so happens, brings the baseball down to a level below the level of where the sheet is being held. The 'curvature' that the baseball's weight causes in sinking down into the latex sheet, in turn, is an accurate portrayal of the 'curvature' caused by mass in reality. Having been introduced to the concepts of mass and curvature, we must address the main issue: what does it mean for mass to curve space-time?

We are to think of space-time - our concept of the universe - as being an expanding elastic surface. As expressed by the theory of relativity, an expanding elastic surface is in many ways an accurate description of what a universe is. Why view space-time in this manner? According to the theory of relativity, the two primary influences active within our universe are gravity and time. In giving meaning to these concepts, the theory of relativity needed to convey a medium within which gravity and time could operate. That medium turned out to be an 'expanding elastic surface'. Representing space-time in this manner was a wise choice, for as it can be seen thinking of space-time in this manner succeeds in explaining the concepts we know as gravity and time. As the means of conveying these concepts, we will conduct an experiment. In this experiment we 'construct our own universe', from scratch. As the first step in doing so we are to recall that every universe requires the presence of 3 basic components in order to exist. These components consist of an elastic surface, motion, and mass. Gravity and time are products of these 3 components. Gravity and time, furthermore, are in many ways the entities without which the universe as we experience it would not exist.

As you may recall, mass curves space-time. What this means, given the content of the statement, is that when mass curves the elastic space-time surface, it affects gravity and time equally, because the surface is a union of space and time. Let us take this a step further. The curvature that mass brings about upon the space-time surface, it is agreed, affects both gravity and time. Things do not stop there, however. When we take a closer look, we find that the curvature that mass brings about upon the space-time surface not only affects gravity and time, but is the source and origin of all gravity and time. What this means, quite simply, is that wherever mass is, gravity and time will also be. Given the reasoning of this argument, furthermore, we can conclude that because the curvature caused by mass is, once again, the source and origin of all gravity and time, gravity and time require the presence of mass to exist. For gravity and time to exist apart from the influence of mass, in turn, is an impossibility. This makes perfect sense: since when can gravity take place, without a massive body through which the gravity makes itself known? Or how can time cause aging to occur, without the presence of aging entities through which the effects of time can be observed?

Exactly how, then, does 'curvature of surface', as previously described, produce the effects we call gravity and time? As you may recall, the ingredients required to make a universe are an elastic surface, motion, and mass. Let us focus our attention upon the 'motion'. The 'motion' being spoken of in the statement just presented is, of course, motion of the elastic surface. For ease of visualization we will think of the moving elastic surface as being the surface of an expanding elastic sphere. To give added meaning to the material, the outward expansion of the elastic sphere will be referred to as space-time pull, in the sense that a surface in motion is a surface being 'pulled' - pulled in a way that has a direct effect on space-time. What is gravity? What is time? So far we know that they are the 'curvature' of the space-time surface, and that gravity and time are affected equally by surface curvature. Not until now has the situation been adequately suitable to define just what gravity and time are. Gravity and time are, quite clearly, the results of a surface's state of motion acting upon that surface's curvature. The motion, of course, exists in the form of space-time pull. The surface curvature, in turn, refers to the space-time surface curvature caused by mass. It is the motion of the space-time pull that 'breathes life' into the surface curvature, yielding what we call gravity and time. Exactly what is the nature, you may ask, of this 'action of motion' upon surface curvature?

As you may recall, the surface we are working with is an elastic surface. An elastic surface, by nature, can be altered in many ways. In the material to come we will become more familiar with gravity and time by studying the ways in which an elastic surface can be altered. In being made aware of how an elastic surface can be altered, in turn, we produce the means by which we can see the fundamental mechanisms that lie behind gravity and time. What we know, once again, is that gravity and time are similar in the sense that they are both results of a surface's state of motion acting upon that surface's curvature. The relatedness of gravity and time, furthermore, can be considered to be reflected in the fact, as you may recall, that surface curvature affects gravity and time equally. What we don't know, however, is what makes gravity and time different. As the means of better understanding this, we must put into consideration that when an elastic surface is altered, many properties of the surface are affected. Of the possible properties an elastic surface can take on, 2 are of importance to us. The first property of elastic surface being spoken of is the elastic surface's physical shape. The second property, in turn, is the stretching of fabric associated with the elastic surface. Each property portraying an altered elastic surface corresponds, take note, to its own space-time process. The elastic surface property we know as physical shape applies to the space-time process we know as gravity. The elastic surface property we know as stretching of fabric, in turn, applies to the space-time process we know as time.

As the means of expressing how each property of elastic surface is related to the space-time process that corresponds to it, we will build a model representing a region of curved space-time surface. As we had done earlier, we are to think of an outstretched thin latex sheet, held at its edges. Next, we obtain a coin, and then rest the coin flatly onto the center of the outretched thin latex sheet. We then place our thumb firmly against the coin, pushing down hard so that the coin sinks down deeply into the latex sheet. The coin, in curving the surface of the latex sheet, represents a 2-dimensional planet whose mass, as demonstrated by the force being exerted upon the coin, curves the space-time surface surrounding it. What we know, first of all, is that the elastic surface surrounding the coin has a documentable physical shape. Equally so, when the coin is pressed down into the elastic surface surrounding the coin, the coin 'sinks down' into that surface, producing a distinct stretching of fabric. What has yet to be added to our model? In order to make a model that can simulate the space-time processes of gravity and time, our model requires all of the ingredients required to make a universe: an elastic surface, motion, and mass. The latex sheet qualifies as an elastic surface. The coin, in turn, qualifies as mass. We have not yet, however, added motion to our model - motion we would understand the latex sheet to be engaged in. This motion, to use the term introduced earlier, is to be thought of as space-time pull. In what way could we indicate that the latex sheet in our model is in motion? It is quite clear, first of all, that visualizing the constant state of motion that the model requires is not an easy task - it is hard to keep up with a model that is in constant motion. Is there a way around this?

Yes. There is a way in which we can simulate the effects of motion - the space-time pull - and yet avoid the obstacle of a constantly moving model. We do this by adding a fan to the model. The fan is to be placed up above the center of the latex sheet, aimed down at the coin at the bottom of the depression below. What, then, is the fan's purpose? The fan's purpose is to simulate the conditions that we would expect the latex sheet to undergo were it in motion. How, then, does this work? Grasping the concept behind the system at work here is in no way difficult. When a surface is engaged in a state of motion, a certain force is applied onto the surface, as a result of the motion. This force 'presses down' upon the surface. In aiming a fan down at a stationary surface we are to imagine the current from the fan to 'press down' upon the surface in the same way the surface would be 'pressed down' upon by being in motion. As a result, we can easily envision the effects of a surface engaged in a state of motion, without being required to visualize the actual motion involved! By definition, then, the fan in this model represents what we have come to know as space-time pull - the influence that 'breathes life' into space-time surface curvature, yielding what we call gravity and time.

How does space-time pull 'breathe life' into the curvature of the surface it carries along? Having constructed our model of a curved space-time surface, we now possess the ability to physically envision the processes of gravity and time - concepts that up until now we have relied mainly upon words to express. Given this opportunity, we are once again to see the latex sheet in our model for what it is - a union of space and time. When the surface is curved, gravity and time are affected to equal extents. Why? How do 2 separate processes originate from a common curvature? Not until now have we possessed the ability to explain exactly why space-time surface curvature distributes its influence to gravity and time equally. Apparently gravity and time are linked at a very fundamental level. To understand how, we must see gravity and time for what they are: the results of a surface's state of motion acting upon that surface's curvature. In doing so, when we think of gravity and time we are to think of the property of elastic surface associated with each.

The physical shape of an elastic surface is attributed to gravity. To understand this, we will refer to our model of curved space-time. The coin in the model, a 2-dimensional planet, lies below the level of where the latex sheet is being held, at the bottom of a downhill depression. We will refer to the downhill region of the latex sheet that surrounds the coin as 'curved space'. The part of the latex sheet unaffected by the depression caused by the coin will be referred to as 'flat space'. 'Flat space' is associated with weightlessness. 'Curved space' is associated with gravity.

To understand this, imagine placing a button down upon a region of 'flat space' on the surface of the latex sheet. We are to recall that the latex sheet of our model is understood to be in a state of motion - under the effects of space-time pull. We are to further recall that our means of handling surfaces in motion is to have the surface remain stationary and to hold a fan up above the surface, imagining the current from the fan as the equivalent of the force that the surface, were it in motion, would apply. Observe, if you will, that as the fan sends its current down upon the button, the button does not move - it is 'weightless'. This is because the button is in 'flat' space and not 'curved' space. As the next part of the experiment, we turn off the fan up above the latex sheet. We then place the button on the slanted slope making up the middle region of the depression that surrounds the coin. With no force to push it forward, the button does not move. When we turn the fan back on, however, an interesting thing happens: the button moves toward the coin! The button, in being 'pushed' into the depression, in response to that push "slides" downhill toward the coin. We associate this downhill "slide" with gravitational pull. The process being described is space-time pull at work: a surface's state of motion acting upon that surface's curvature. It is in a manner very similar to the process just described that the physical shape of an elastic surface brings about the effects of gravity. In what way, then, can we consider the stretching of fabric of an elastic surface to bring about activity related to time?

In the model described above, we witnessed how space-time pull 'breathes life' into gravity by providing the "push" needed for gravitational pull to occur: mass, in curving the physical shape of the surface, provides the means by which space-time pull acts upon that curvature, yielding gravity. Now we are presented with the challenge to use our model of curved space-time to express how space-time pull 'breathes life' into time. We are familiar with the notion that mass produces gravity. We have no difficulty in agreeing that mass produces gravity - the earth produces gravity; the sun produces gravity; moons produce gravity. That mass produces gravity is a valid statement, in the sense that mass is the source of the surface curvature that makes gravity possible. It is common practice, furthermore, to associate the mass that produces gravity, with the massive body we understand the mass to belong to. In order to continue forth in the material, we must take this notion a step further: we are to be made familiar with the idea that mass produces time. This at first may seem like a strange concept. It is strange, in fact, in comparison to the means of reasoning we live by. Once grasped, however, it becomes as straightforward as the relationship we are all aware of that exists between mass and gravity. In fact, it is by understanding how gravity relates to mass that we can in turn understand how mass is related to time. How do we give meaning to an occurrence as unusual as mass producing time?

Doing so involves asking a very deep, very pressing question: what is time? In response to this question we are to address the situation with both insight and logic. We will ask ourselves the following question: what is it we will always find whenever time is present, yet will never find in time's absence? The answer: objects! Our only evidence for the existence of time consists of the sensory data we obtain from aging physical objects. What this means is that the only reason time exists is because it must make itself known to live up to the time-related demands of aging physical objects. Therefore, time only exists when needed: time can exist only when objects exist. This makes sense in terms of logic and reason. However, in the physical, empirical world, this is scientific fact: time can exist only when objects exist. In order to understand this proposal we must be willing to approach time from an entirely new point of view: the view that mass produces time.

Let us, for the sake of the topic we are discussing, recall an earlier topic and consider the following question: in what way is time the action of space-time pull upon surface curvature? In review: mass produces time. Time is a product of mass. It is a physical, empirical fact that time cannot exist apart from mass. What makes these statements true? Let us return to our model of curved space-time. We are once again to turn on the fan up above the latex sheet. Thinking of the fan as 'pressing down' against the surface of the latex sheet, as you may recall, allows us to envision the way the surface would be 'pressed down' upon were it in motion - yet we need visualize nothing moving. The latex sheet, then, is to be considered to be undergoing the effects of space-time pull. We are to imagine, therefore, the current from the fan to bring about the continuous entry of certain random, scattered point-like entities directly into all regions of the surface of the latex sheet. We will refer to these point-like objects as "potential events". These "potential events" are, as stated, point-like. By definition, "potential events" are events that lack both order and separation by time - qualities we understand all events to possess. To imagine such an entity, imagine the events of all of time to occur in a single instant. Consider, afterward, the nature of the arrangement just described. As can be clearly seen, no event is distinct from another. The inevitable result to this, furthermore, is impossibility of designating sequence among the events. Let us move on.

What is time, and how is it produced by mass? The basic unit of time is the event. We think of time by thinking in terms of events. Perhaps our most straightforward understanding of the concept of time, furthermore, lies in perceiving events as they appear without time. Without time, there exists no means of designating one event from another - no means exists of specifying sequence or order of event. If an event exists under the conditions of what we understand to be within the boundaries of time, then, what would be different? An event that lies within the boundaries of time is an event that has claimed a definite separateness from other events, and possesses within that array of events a unique linear location. These conditions are quite evident. What, however, makes the random, scattered "potential events" entering into all regions of the latex sheet what they are? What makes the random, scattered "potential events" entering into all regions of the latex sheet what they are is that they lack time. What role does mass play here?

How does mass express the concept of time to all of the incoming "potential events", given that they apparently lack time? We must remind ourselves as to what time is: the action of space-time pull upon surface curvature. What this means is that time can only occur in the presence of mass. Why? Recall, if you will, our encounter with gravity performed by means of our model of curved space-time. The situation involved 'curved space' and 'flat space'. 'Curved space' is the equivalent of the surface of the latex sheet affected by the downhill depression surrounding the coin. 'Flat space' is the part of the surface of the latex sheet unaffected by the coin. The button we introduced into the model served as our conception of an object being acted upon by gravitational forces. When acted upon by space-time pull (the current from the fan), the button took the straightest possible path it could take based upon the presence or absence of surface curvature. In 'flat space' the straightest possible path was not to move at all. In 'curved space', the straightest possible path was to move toward the coin. "Potential events" entering into the surface of the latex sheet all take the straightest possible path possible as well. The straightest possible path for "potential events" entering into flat space is not to move at all. Their journey ends there, and no time takes place. For "potential events" entering into 'curved space', however, the straightest possible path is directly downhill, toward the center of the depression surrounding the coin. This motion is the motion we associate with the passage of time. Given this, it is quite clear, then, that time will not occur in a universe lacking objects, because the surface curvature that is a result of the presence of objects is the very curvature that makes time possible. After considering this, take note, afterward, that the placing of the "potential events" onto their designated paths is but the first half of the task mass assumes in producing time: mass has yet to express the concept of time to the incoming "potential events" - entities that, as stated, lack time.

In review: gravity and time are the two primary influences active within our universe. Space-time - our concept of the universe - can be thought of as an expanding elastic surface. The presence of mass within that surface, furthermore, brings about what we call curvature of that surface. Space-time pull - the motion we understand an expanding elastic surface to be engaged in - acts upon this surface curvature, 'breathing life' into it, yielding what we call gravity and time. Different degrees of surface curvature, take note, are possible. However, the presence of surface curvature will always affect gravity and time equally, because the surface being curved is a union of space and time. How, then, do two separate processes result from but a single curvature? Two processes can result from but a single curvature in the sense that different aspects of that curvature are acted upon. When mass curves the elastic space-time surface, two different properties of that elastic surface are altered: the surface's physical shape and the stretching of fabric of the surface. The physical shape of the surface applies to gravity: objects receiving the "push" given by space-time pull follow the straightest possible path that surface curvature provides - a path of no motion at all (weightlessness) or a direct path forward (gravitational pull). The stretching of fabric of a surface (due to the mass present), in turn, is the property of elastic surface that applies to time - the property of elastic surface that we have yet to cover and will cover right now.

In what way can the stretching of the fabric of an elastic surface tell us how mass produces time? Let us return to our model of curved space-time. Space-time pull (the current from the fan positioned up above the latex sheet and aimed down at it), is drawing into the latex sheet an assortment of certain random, scattered point-like "potential events" - events that as stated lack both order and separation by time. Let us move on. There are several vital questions before us. Among them: how does mass express the concept of time to entities that apparently lack time? How does space-time pull 'breathe life' into space-time curvature, yielding what we call time? How does mass produce time? We need only familiarize ourselves with the basic attributes of a stretched elastic surface: the fabric of an elastic surface is elongated when stretched. When stretched, furthermore, the fabric of an elastic surface is spread out. What happens when the "potential events" enter into the surface? "Potential events" that enter into the latex surface in flat space meet the end of their journey there: the straightest possible path that can be taken within the latex surface upon arrival is not to move at all. No time is produced.

The straightest possible path for 'potential events' that enter into the latex surface in 'curved space', however, is directly downhill, toward the center of the depression surrounding the coin. As stated, this motion is motion to be attributed to the motion we associated with the passage of time. It is but the first half of the task mass assumes in producing time. What remains to be done by mass is to express the concept of time to the incoming "potential events" - entities that, as stated, lack time. When "potential events" enter into the space-time surface curvature surrounding an instance of mass, an intricate process takes place. The physical configuration of the stretched fabric of the surface "stretches out" the incoming "potential events", placing distance in between events that beforehand had no means of designating separateness from each other. 'Pulling apart' the "potential events", in turn, assigns sequence and order to the events - attributes we understand the typical event to possess. The passage of time is given meaning in the sense that delay now exists in between events - events do not happen 'all at once', so to speak. Events, rather, possess a distinct order and sequence to the manner in which they occur. Having engaged in these intense studies of gravity and time, what can be concluded? Perhaps the concept most applicable to the situation is that of how our everyday experiences of gravity and time hide from us the details of the actual mechanism behind their operation. We experience gravity as a 'pulling force'; we experience time as change. Yet we cannot directly see gravity and time for what they actually are: actions upon an elastic surface. The same is true for the 4 seasons of the year: what we experience as changes in temperature and as celebration of holidays actually happens to be a skillful tilting of the earth's axis.