We will now discuss the changes in the notion of time and space brought about by the development of the special theory of relativity. Let us start with a brief characteristic of the classical account of space and time as incorporated in the Galilean-invariant version of Newtonian mechanics (i.e. the version that dispenses with the concept of absolute motion and absolute position). The main assumption is that no inertial frame of reference is privileged, and uniform motion is relative. However, the notion of simultaneity remains absolute, i.e. frame-independent. For each moment of time the set of events occurring at that moment defines a three-dimensional space with the usual Euclidean metric (distance) attached to it. Hence, Galilean space-time foliates naturally into separate spaces defined at different times. However, there is no absolute connection between points in spaces at different times (no absolute co-location). The question of which point of space at t2 is a continuation of a point at t1 does not receive a frame-independent answer. If we think that spatial points (places) are objects which retain their identity over time, then no such objects are present in the Galilean version of classical mechanics.
A step towards special relativity is the realisation that the relation of simultaneity has no obvious empirical content, due to the fact that all signals (including light) travel at finite speeds. What we observe as our ‘present’ is actually already in the past (the farther, the more distant the observed event is). The standard, operationally defined notion of simultaneity is given as follows: two events x and y are simultaneous iff light signals sent from x and y meet exactly half way between x and y. But this definition is obviously not frame-independent. Suppose that the definition of simultaneity is satisfied in a frame f, and let us consider another frame f’ which moves with respect to f in the direction of the event y. The spatiotemporal point where the two beams of light meet will not be located in the middle of the distance between their locations x’ and y’, but rather closer to x', so from the perspective of f’ y happened earlier than x. We have to add that the signal definition of simultaneity presupposes that the speed of light is constant in all frames of reference.
In special relativity neither simultaneity nor co-location are invariant notions (independent of the frame of reference). Thus space-time cannot be absolutely divided into space and time. However, there is a relation between events (spatiotemporal points) which stays the same in all frames of reference. This relation is defined by the so-called spatiotemporal interval: cdt^2 – dx^2 – dy^2 – dz^2., where c – the speed of light, and dt, dx, dy and dz are temporal and spatial intervals between the two events. Two events for which the spatiotemporal interval is positive are called ‘time-like separated’. Such events can be connected by a signal travelling slower than light. If the interval equals zero, the events can be only connected by a beam of light. Events separated by a negative interval are called space-like separated. Such events cannot directly communicate by way of sending signals.
The basic structure of relativistic space-time (so-called Minkowski space-time) can be given with the help of light cones. For a given event x, its forward light cone consists of all events reachable from x by beams of light. Similarly, x’s backward light cone contains all events which can reach x using beams of light. The area within x’s backward light cone is called its absolute past, and within the forward light cone its absolute future. The events outside of both light cones are neither past nor future with respect to x, but they cannot be interpreted as being simultaneous with x. Their temporal relation with x is frame-dependent: for every event y space-like separated from x there is a frame of reference f in which x and y are simultaneous. But if we choose a different event y’ also space-like separated from x, the frame of reference in which y’ is simultaneous with x will be generally different from f.
Now we will discuss some issues related to the asymmetry of time. Let us start with the problem of time travel. Is time travel conceptually possible, or does it involve logical contradiction? First we have to decide what process can be called time travel. For a given object we can say that it travels in time if there is a difference between its own time and the external time of the world. If the interval measured with the object’s time is shorter than the external interval, we can speak about travel into the future. Such travel is not only possible but actually happens, according to the special theory of relativity. Due to the effect known as time dilation, if an object moves, its own time measures shorter intervals than the external time. If a traveller embarks on a journey and then comes back, his clock will show that his journey was shorter than when measured by the external clocks (this is the basis of the so-called twin paradox).
The most radical type of time travel is when the traveller goes into the past, i.e. the duration of his journey measured according to the external time is actually negative. Some philosophers claim that travel into the past involves contradiction, because a time traveller could change the past, and this is impossible. More specifically, the concept of changing the past is applied to states of affairs. In order to change a given past state of affairs – for instance, by scratching an inscription on a rock a thousand years ago – this state of affairs (the rock being unscratched) has to be both real (‘before’ the change) and unreal (‘after’ the change). But of course the time of the occurrence of these two contradictory states of affairs is the same, so the contradiction seems unavoidable. But it can be observed that the same problem applies to the apparently uncontroversial case of a change in the future. To literally change a state of affairs at a future time t requires that this state of affairs exist before but not after my action. But, again, this leads to logical contradiction. This difficulty can be avoided, though, when we apply the notion of change to things, not states of affairs. I certainly changed the past rock: before my intervention it wasn’t scratched, and afterwards it bore an inscription.
The most celebrated grandfather paradox exploits the possibility of vicious causal loops that seems to be opened by admitting time travel. The time traveller goes back to the times of his grandfather’s youth and kills him in the past. If his grandfather dies before he can have any children, the traveller will not be born in the future, and a contradiction ensues: the traveller both exists in the future (because he came from there to kill the grandfather) and does not exist (because his grandfather dies childless). This paradox has the following general form: there are two events A (the beginning of the journey to the past) and B (the killing of the grandfather) such that A is later than B, A causes B and B causes non-A. In order to avoid the problem, serious restrictions have to be imposed on the possible interactions of the time traveller with the past. Speaking loosely, each time the traveller tries to kill his grandfather something must get in the way to prevent him from accomplishing this task. Note that this extends to any action of the traveller (even seemingly innocent, such as leaving footprints on the grass) which might lead to the consequences threatening the entire future travel to the past as it precisely occurred. One general solution may be to assume that causal links directed from the future to the past do not ‘couple’ with the causal links leading in the ordinary temporal direction. But this would effectively imply that the traveller could only observe the past and not interact with it the normal way.
Reading:
B. Garrett, "Time travel", pp. 94-99, What is this thing called metaphysics?
Showing posts with label simultaneity. Show all posts
Showing posts with label simultaneity. Show all posts
Saturday, March 27, 2010
Wednesday, February 24, 2010
Temporal relations
We will start our analysis of metaphysical aspects of time with temporal relations. Temporal relations can be defined on both things and events, but because things vary greatly with respect to their duration, it is more convenient to choose events instead. Another simplifying assumption we have to make is that we will consider events as being momentary (point-like): they are assumed to occupy a single moment in time, not an interval (no duration). The main temporal relation is the earlier-than relation E. Julius Caesar’s death is earlier than the fall of Constantinople, and the fall of Constantinople is earlier than the storming of the Bastille. The main formal features of the earlier-then relation are as follows: it is irreflexive (no event is earlier than itself), asymmetric (if x is earlier than y, then y is not earlier than x), and transitive. These three characteristics ensure that the relation of being earlier is a strong ordering of the set of all events. To this we will add the requirement of linearity, which can be explicated as follows. First let us define the following relation R: R(x, y) iff neither E(x, y), nor E(y, x) (neither x is earlier than y, nor y is earlier than x). The requirement of linearity of the relation E amounts to the condition that R be a relation of equivalence, i.e. reflexive, symmetric and transitive. If that is the case, we can call R the relation of simultaneity. An example of a situation when E is not linear is a so-called branching time. If events are ordered on a tree with different branches pointing towards the future, the conditions of irreflexivity, asymmetricity and transitivity of E are satisfied, but the linearity is violated, for events located on different branches are not comparable (one is neither earlier than, nor later than, nor simultaneous with the other one). More precisely, the condition of the transitivity of the relation R is violated in this case, since we can choose two events x and z on one branch and a third one y on another branch, from which it clearly follows that R(x, y) and R(y, z), but not R(x, z). Some philosophers insist that the set of all events should be given the structure of a tree due to indeterminism, but we will continue to assume the linearity condition.
The relation of simultaneity can help us define an important category of temporal objects: moments (temporal instants). A moment at which an event x occur is just the set of all events simultaneous with x. Moments are equivalence classes of events with respect to simultaneity. Notice that the ordering relation “earlier than” between events can be naturally extended for moments. Moment m is earlier than moment n iff all events participating in m are earlier than the events constituting n. This definition is formally correct, because the relation E is invariant with respect to substitution of simultaneous events. One consequence of the proposed definition of moments is that there can be no empty moments (instants with no events). This has serious consequences for the controversy between two ontological positions regarding the nature of time: absolutism (substantivalism) and relationism. We will discuss this problem later.
Another issue is the problem of time measurement. How to measure the duration of an interval between two events x and y? One possibility is to use a cyclic (repeatable) process, for instance a pendulum, a water clock, or a sundial, and to count how many cycles happen during the interval from x to y. But the main problem of this approach is how to make sure that the subsequent cycles have the same durations. If we use another cyclic process to prove that our initial measuring device was uniform, we will end up in a regress. A solution is to adopt a conventionalist answer to the question of how to compare lengths of different time interval. But accepting conventions does not imply total arbitrariness. We should adopt conventions which make sure that the fundamental laws of physics involving time (such as the laws of Newtonian mechanics) have the simplest possible mathematical form.
Some philosophers insist that there is one important aspect of time which is missing from our analysis so far. It is the dynamical aspect of time expressed in the division of events into three spheres: past, present and future. The passage of time is not included in the earlier-than relation. It can be expressed in the observation that events move from being future into being present, and finally they turn into past events. John McTaggart introduced a fundamental distinction, to this day referred to in virtually all publications on the subject of time. It is the distinction between the A series and the B series. The B series is the set of all events together with the relation “earlier than”, while the A series is the set of all events divided into past events, present events and future events. An important difference between these two interpretations of time is that the B interpretation can be given exclusively in a tenseless language, while the A theory requires the use of tensed forms of verbs. For instance it is correct to say that the Battle of Hastings is earlier that the Battle of Waterloo, where the verb “be” has an atemporal sense, not relativised to the present moment. In contrast, the Battle of Hastings is now past, but it was present and had been future, while my current lecture is present, was future and will be past. For McTaggart this fact shows that the B series is static, “frozen in time”, eternal, while the A series is dynamic, moving, changing. We should also note that the A series descriptions cannot be definitionally reduced to the B type expressions. Using the relation of being earlier than we can define the notions of past present and future only relatively to a given event (moment). The past of an event x is the set of all events earlier than x; the present of x is the set of all events simultaneous with x, and the future of x is the set of all events such that x happens earlier from them. On the other hand, the task of reducing the B-series to the A-series has greater chances of success. For instance, we could try to give the following reductive definition of the relation E: x is earlier than y iff there is a moment of time at which x is present and y is future.
McTaggart makes two significant claims regarding the two approaches to the concept of time. One is that the existence of the A series is necessary for time to exist, and therefore for the existence of the B series as well. The other claim is more radical: McTaggart insists that the A series is contradictory. From these two claims it follows that time does not exist. McTaggart fully embraces this consequence. McTaggart argues in support of the first claim as follows. His main point is that the A series is necessary in order to express the notion of change. For McTaggart the notion of change applies to events only: each event changes from being in the far future to being in the closer future, then to being present, and then to being past. But an objection can be raised that there is a legitimate notion of change which is applicable to things, not events, and which can be expressed in the B-theory. B. Russell used the following example: a poker put in a fireplace changes from being cold at t1 to being hot at t2. More generally, a change is the fact that a given sentences about an object is true at t1 and false at a later time t2. This interpretation of change does not require the A-series. But McTaggart retorts that the change described in Russell’s example is spurious. In fact there is no real change here at all, since it is always true that the poker is cold ad t1 and hot at t2. To bolster his claim, McTaggart uses an argument from analogy. Consider the zero meridian and two points on it: one m1 in England and one m2 in France. The meridian can be ordered in the same way events (moments) are ordered in the B-series. But now the sentence “This point on the meridian lies in England” is true at t1 but false at t2. So, according to Russell’s definition, there is a change happening here. But clearly we see that nothing really changes, hence Russell’s definition is incorrect.
In response to McTaggart argument it can be pointed out that the analogy he uses is incomplete and therefore it does not warrant the conclusion. First, the meridian case lacks a counterpart of the poker in Russell’s example: a thing that retains its identity in spite of the change in properties. The statement which is supposed to be true at m1 but false at m2 is not about any thing which exists at both points. One may try to correct McTaggart’s argument in the following way: suppose that the required counterpart of the thing in the temporal case is the entire Great Britain, and the sentence considered is “Great Britain is sunny”, which happens to be true at some point on the meridian but false at a different point. But even here the analogy is not sufficiently strong: we couldn’t claim by any stretch of the imagination that the whole Great Britain is wholly present at any single point on the meridian, whereas it is typically assumed that things are wholly present at temporal instants. The second objection to McTaggart’s argument is that it arbitrarily selects one method of ordering points on the meridian (either from the South Pole to the North Pole or vice versa), whereas the temporal case has an objective temporal direction independent of our decision.
Readings:
B. Garrett, "Time: The fundamental issue", pp. 69-82, What is this thing called metaphysics?
The relation of simultaneity can help us define an important category of temporal objects: moments (temporal instants). A moment at which an event x occur is just the set of all events simultaneous with x. Moments are equivalence classes of events with respect to simultaneity. Notice that the ordering relation “earlier than” between events can be naturally extended for moments. Moment m is earlier than moment n iff all events participating in m are earlier than the events constituting n. This definition is formally correct, because the relation E is invariant with respect to substitution of simultaneous events. One consequence of the proposed definition of moments is that there can be no empty moments (instants with no events). This has serious consequences for the controversy between two ontological positions regarding the nature of time: absolutism (substantivalism) and relationism. We will discuss this problem later.
Another issue is the problem of time measurement. How to measure the duration of an interval between two events x and y? One possibility is to use a cyclic (repeatable) process, for instance a pendulum, a water clock, or a sundial, and to count how many cycles happen during the interval from x to y. But the main problem of this approach is how to make sure that the subsequent cycles have the same durations. If we use another cyclic process to prove that our initial measuring device was uniform, we will end up in a regress. A solution is to adopt a conventionalist answer to the question of how to compare lengths of different time interval. But accepting conventions does not imply total arbitrariness. We should adopt conventions which make sure that the fundamental laws of physics involving time (such as the laws of Newtonian mechanics) have the simplest possible mathematical form.
Some philosophers insist that there is one important aspect of time which is missing from our analysis so far. It is the dynamical aspect of time expressed in the division of events into three spheres: past, present and future. The passage of time is not included in the earlier-than relation. It can be expressed in the observation that events move from being future into being present, and finally they turn into past events. John McTaggart introduced a fundamental distinction, to this day referred to in virtually all publications on the subject of time. It is the distinction between the A series and the B series. The B series is the set of all events together with the relation “earlier than”, while the A series is the set of all events divided into past events, present events and future events. An important difference between these two interpretations of time is that the B interpretation can be given exclusively in a tenseless language, while the A theory requires the use of tensed forms of verbs. For instance it is correct to say that the Battle of Hastings is earlier that the Battle of Waterloo, where the verb “be” has an atemporal sense, not relativised to the present moment. In contrast, the Battle of Hastings is now past, but it was present and had been future, while my current lecture is present, was future and will be past. For McTaggart this fact shows that the B series is static, “frozen in time”, eternal, while the A series is dynamic, moving, changing. We should also note that the A series descriptions cannot be definitionally reduced to the B type expressions. Using the relation of being earlier than we can define the notions of past present and future only relatively to a given event (moment). The past of an event x is the set of all events earlier than x; the present of x is the set of all events simultaneous with x, and the future of x is the set of all events such that x happens earlier from them. On the other hand, the task of reducing the B-series to the A-series has greater chances of success. For instance, we could try to give the following reductive definition of the relation E: x is earlier than y iff there is a moment of time at which x is present and y is future.
McTaggart makes two significant claims regarding the two approaches to the concept of time. One is that the existence of the A series is necessary for time to exist, and therefore for the existence of the B series as well. The other claim is more radical: McTaggart insists that the A series is contradictory. From these two claims it follows that time does not exist. McTaggart fully embraces this consequence. McTaggart argues in support of the first claim as follows. His main point is that the A series is necessary in order to express the notion of change. For McTaggart the notion of change applies to events only: each event changes from being in the far future to being in the closer future, then to being present, and then to being past. But an objection can be raised that there is a legitimate notion of change which is applicable to things, not events, and which can be expressed in the B-theory. B. Russell used the following example: a poker put in a fireplace changes from being cold at t1 to being hot at t2. More generally, a change is the fact that a given sentences about an object is true at t1 and false at a later time t2. This interpretation of change does not require the A-series. But McTaggart retorts that the change described in Russell’s example is spurious. In fact there is no real change here at all, since it is always true that the poker is cold ad t1 and hot at t2. To bolster his claim, McTaggart uses an argument from analogy. Consider the zero meridian and two points on it: one m1 in England and one m2 in France. The meridian can be ordered in the same way events (moments) are ordered in the B-series. But now the sentence “This point on the meridian lies in England” is true at t1 but false at t2. So, according to Russell’s definition, there is a change happening here. But clearly we see that nothing really changes, hence Russell’s definition is incorrect.
In response to McTaggart argument it can be pointed out that the analogy he uses is incomplete and therefore it does not warrant the conclusion. First, the meridian case lacks a counterpart of the poker in Russell’s example: a thing that retains its identity in spite of the change in properties. The statement which is supposed to be true at m1 but false at m2 is not about any thing which exists at both points. One may try to correct McTaggart’s argument in the following way: suppose that the required counterpart of the thing in the temporal case is the entire Great Britain, and the sentence considered is “Great Britain is sunny”, which happens to be true at some point on the meridian but false at a different point. But even here the analogy is not sufficiently strong: we couldn’t claim by any stretch of the imagination that the whole Great Britain is wholly present at any single point on the meridian, whereas it is typically assumed that things are wholly present at temporal instants. The second objection to McTaggart’s argument is that it arbitrarily selects one method of ordering points on the meridian (either from the South Pole to the North Pole or vice versa), whereas the temporal case has an objective temporal direction independent of our decision.
Readings:
B. Garrett, "Time: The fundamental issue", pp. 69-82, What is this thing called metaphysics?
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