Thus, the hole argument fails.
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I would apply the term generally covariant to the conclusion that an entire equivalence class of carry alongs corresponds to one physical solution. The coordinate-independent versions of all three concepts are obtained by substituting basis vectors for coordinates and diffeomorphisms for coordinate transformations. Our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points.
Moreover, the results of our measurements are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock-dial, and observed point-events happening at the same place at the same time.
The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. We allot to the universe four space-time variables in such a way that for every point-event there is a corresponding system of values of the variables. To two coincident point-events there corresponds one system of values of the variables , i. If, in place of the variables , we introduce functions of them, , as a new system of co-ordinates, so that the systems of values are made to correspond to one another without ambiguity, the equality of all four co-ordinates in the new system will also serve as an expression for the space-time coincidence of the two point-events.
As all our physical experience can be ultimately reduced to such coincidences, there is no immediate reason for preferring certain systems of coordinates to others, that is to say, we arrive at the requirement of general covariance Einstein, , pp. In this version of the argument, there is no mention of dynamical equations or even of fields. Indeed, he proceeds to illustrate it with a version of the trivial identity applied to a system of particles, rather than fields, the model being any set of particle world lines, without any requirement that they satisfy equations of motion.
This difference has occasioned some confusion in recent discussions of the hole argument. My use of the term always involves the field equations. In later years, Einstein explicitly rejected any positivistic interpretation of the point coincidence argument. He wrote to Schlick:. Generally considered, your presentation of the [point coincidence] argument does not correspond with my conception of it since I find your entire conception too positivistic, so to speak. Physics indeed provides relations between sense experiences, but only indirectly.
For me, its essence is by no means exhaustively characterized by this assertion. Physics is an attempt at the conceptual construction of a model of the real world, as well as of its lawful structure. Indeed it must represent exactly the empirical relations between the sense experiences that are accessible to us; but only in this way is it linked to the latter Einstein to Moritz Schlick, 28 November ; cited from Engler and Renn, , p. In , even before Einstein completed the general theory of relativity, Erich Kretschmann a , b had undertaken an investigation that led him to a version of the trivial identity.
Einstein concedes the point, but argued, not very successfully, 18 that an added criterion of simplicity gives the principle a heuristic significance. Evidently, he was not himself clear on the difference between his two arguments: While the public point coincidence argument does not provide a criterion for singling out theories, the criterion of general covariance in the private argument does. Apparently unaware of Kretschmann , Arthur Komar also suggested the use of four invariants of the Riemann tensor as coordinates.
In subsequent discussions of the problem of true observables in general relativity, they are often referred to as Kretschmann—Komar coordinates. Stachel , noted their use as a way of individuating the points of space-time, and they have subsequently figured in many discussions of the hole argument. David Hilbert, the renowned mathematician, became interested in the problem of a unified gravitational-electromagnetic theory and followed Einstein in arguing against generally covariant field equations. Instead of a hole, however, he formulated the argument in a mathematically more sophisticated way, using a space-like hypersurface.
After Einstein returned to generally covariant field equations, Hilbert dropped this argument against them, and Hilbert is the first discussion of the Cauchy problem in general relativity; but the analysis is far from complete. The monograph version Darmois, …became my bedside reading. In this book …is the first rigorous analysis of the hyperbolic nature of the Einstein equations, i.
Many current discussions of the non-uniqueness problem in general relativity are formulated in terms of the Cauchy problem rather than the original hole argument see, e. RSS News Contacts. Hide Subsections. Abstract 1 Introduction 1. Relativity , 17 , 1, doi This work is licensed under a Creative Commons License. Why should we care? Summary: Where we are headed. Outline of the article. Early History. In cases of perceptual experience things appear some ways rather than others to us. We need to explain this.
Why does the penny one sees look elliptical to one as opposed to some other shape? One answer is that there is something of which one is aware which is in fact elliptical. Other philosophers have simply taken the principle to be obvious. How is it supposed to work? On these grounds, the conclusion of the base case stage is supposed to follow.
And then the ultimate conclusion of the argument can be derived from its further premises. But as French and Walters forthcoming argue, this is invalid. One might be aware of the ordinary object as well as the F thing one is aware of. We should be careful to distinguish not being directly aware of the wall from being directly aware of something which is not the wall.
The argument is invalid in conflating these two ideas.
One option for fixing the argument is to introduce what French and Walters call the Exclusion Assumption cf. But whether this assumption is defensible remains to be seen. In the remainder we leave this and the issue of validity aside and consider responses from different theories of experience.
A hallucination is an experience which seems exactly like a veridical perception of an ordinary object but where there is no such object there to be perceived. Like illusions, hallucinations in this sense do not necessarily involve deception. And nor need they be like the real hallucinations suffered by the mentally ill, drug-users or alcoholics. They are rather supposed to be merely possible events: experiences which are indistinguishable for the subject from a genuine perception of an object. For example, suppose one is now having a veridical perception of a snow covered churchyard.
For more on hallucination, see the essays collected in Macpherson and Platchias What this argument shows, if it is successful, is that one is not perceptually aware of ordinary objects in veridical experiences. The conclusion here is not as general as the conclusion of the argument from illusion, but the more general conclusion is surely not far off: for it would be difficult to maintain that though one is not perceptually aware of ordinary objects in veridical experiences, there are other cases of experience where one is perceptually aware of ordinary objects.
So this argument supports if not entails the rejection of our ordinary conception of perceptual experience. Its aim is to show that an aspect of our ordinary conception of perception is deeply problematic, if not incoherent: perceptual experience cannot be what we intuitively think it is. Once again we can view the argument as having a base case A and a spreading step B Snowdon A simply falls out of what hallucinations are supposed to be, and two principles: first, that awareness of an object is a relation to an object, and second, that relations entail the existence of their relata.
For given our principles, if an hallucination as of an ordinary object is to be a mode of awareness of an ordinary object then there must be an ordinary object there for one to perceive. But no such objects are there in hallucinations, therefore, hallucinations are not cases of awareness of ordinary objects. Where the argument from hallucination is controversial is in the spreading step. The spreading step here gets construed in terms of the idea that veridical experiences and hallucinations are essentially the same; mental events of the same fundamental kind Martin It just means that the difference between veridical experience and hallucination is not to be found in their intrinsic natures.
Though it is not plausible to deny the possibility of illusory experiences though we may argue about how best to construe them; Anthony and Kalderon , the claim that subjectively indistinguishable hallucinations are possible is a little more controversial. How do we really know that experiences like this are possible? Austin , for instance, expresses scepticism. One way to answer this—though certainly not the only way—is to appeal to a broad and uncontroversial empirical fact about experience: that it is the upshot or outcome of a causal process linking the organs of perception with the environment, that our experiences are the effects of things going on inside and outside our bodies.
If this is so, then we can understand why hallucinations are a possibility. For any causal chain reaching from a cause C 1 to effect E, there are intermediate causes C 2 , C 3 etc. In this section we will consider the leading theories of experience of the last hundred years. These theories are understood here as responses to the Problem of Perception. There are a number of theories of perception which are not discussed in this entry, either because they are not responses to this specific problem like the causal theory of Grice and Lewis , and Burge or because they require an entire entry of their own like the phenomenology of Husserl —1 and Merleau-Ponty ; see the entry on phenomenology.
As we understand theories of experience, they operate on two levels. On one level they tell us about the nature or structure of experience, on another level they tell us how what is said at the first level bears on grounding or explaining the phenomenal character of experience. Crudely, and with details and qualifications to explored below:.
The Sense-Datum Theory : Level 1: experience is fundamentally a relation to a non-ordinary object; a sense-datum. The Adverbial Theory : Level 1: experience is non-relational and fundamentally a state of mind adverbially modified in a certain way e. The Intentionalist Theory : Level 1: experience is non-relational and fundamentally a matter of representing ordinary objects in certain ways. The Naive Realist Theory : Level 1: experience is fundamentally a relation to ordinary aspects of mind-independent reality.
Here is what we find: The sense-datum theorist rejects our ordinary conception of perceptual experience. The adverbial theorist tries to improve upon the sense-datum theory, and holds on to Awareness. But it is unclear how they can secure Openness. Intentionalists and naive realists hold to both Openness and Awareness , but they do so in different ways, and with different responses to the Problem of Perception.
The way these positions emerge in response to the Problem of Perception is mapped most clearly in Martin , , On the sense-datum theory, a perceptual experience in which something appears F to one consists in a relation of perceptual awareness to something which is actually F Level 1.
So whenever a subject has a sensory experience, there is something of which they are perceptually aware. A sense-datum theorist calls the object of an experience a sense-datum. We can thus re-formulate the Phenomenal Principle espoused by the sense-datum theorists in these terms:. If there sensibly appears to a subject to be something which possesses a particular sensible quality F then there is something—a sense-datum—of which the subject is directly aware which does possess that sensible quality. For the sense-datum theorist, the character of an experience is somehow explained at least in part by the sensible qualities of the sense-datum one is aware of.
Level 2. Consider an experience which one would describe in terms of seeing a snow covered churchyard for what it is. We can isolate certain aspects of the phenomenal character of such an experience, such as the appearance of whiteness to one. We want a theory of experience to explain such aspects.
The sense-datum theorist will claim that things appearing white to you consists in your perceptual awareness of a white sense-datum. The character of your experience is explained by an actual instance of whiteness manifesting itself in experience. For at the moment we have no opposition between sense-data and ordinary objects.
Suppose that one has an experience of a churchyard as described above, and so one is perceptually aware of a white sense-datum. All we know about sense-data is that they must satisfy two conditions:. But paradigm sense-datum theories are, in contrast, non-minimal. And such non-minimal sense-datum theorists do reject our ordinary conception of perceptual experience. We can make sense of this if we move to a less minimal conception of sense-data themselves, on which they are not ordinary aspects of mind-independent reality.
One option here is to articulate a theory of sense-experience and sense-data independently of the problem of perception Jackson and Lowe But by far the most popular approach for sense-datum theorists has been to move to a more committed conception of sense-data on the basis of arguments like the argument from illusion and the argument from hallucination. How does this work?
Take first the argument from illusion. Suppose one sees a white wall as yellow. And suppose the illusion occurs in a world devoid of ordinary objects which are in fact yellow. Thus the sense-datum is a non-ordinary object. Suppose we accept further that in illusions we are directly aware of such non-ordinary sense-data instead of and not as well as the relevant ordinary objects.
So illusions are cases where one is directly aware of just a non-ordinary sense-datum. With this the sense-datum theorist accepts the spreading step and concludes that no perceptual experience is a case of awareness of an ordinary object, and that illusory and veridical experiences are cases of awareness of non-ordinary sense-data.
We can run a similar line of thought with the argument from hallucination. Suppose one has an hallucinatory experience as of a churchyard covered in white snow. But there is no such churchyard there to be perceived. The sense-datum theorist conceives of this experience as a case of perceptual awareness of a white sense-datum. Suppose the hallucination occurs in a world devoid of ordinary white things, so the sense-datum one is aware of is a non-ordinary sense-datum.
The sense-datum theorist then endorses the spreading step stage of argument to get the conclusion that even in veridical experiences one is aware of just non-ordinary sense-data. Initially, the arguments from illusion and hallucination were presented as aiming for a negative claim. That is, in combination:. The arguments, understood as such, are not arguments for the minimal sense-datum theory—the arguments presuppose such a theory.
Rather, they serve as arguments for the transition from the minimal form of the sense-datum theory to the non-minimal form which invokes non-ordinary sense-data. Sense-datum theorists will divide over exactly how to understand sense-data insofar as they are non-ordinary. The early sense-datum theorists like Moore considered sense-data to be mind-independent, but non-physical objects.
Later theories treat sense-data as mind-dependent entities Robinson , and this is how the theory is normally understood in the second half of the twentieth century. What these different workings out of the theory have in common, though, is that they stand opposed to our ordinary conception of perceptual experience in both its aspects. On the non-minimal sense-datum theory, perceptual experiences are presentations not of ordinary objects, but of sense-data, and the character of experience, though dependent on its objects, is thus dependent upon non-ordinary objects. Thus Openness is false. The sense-datum theorist need not deny that we are presented with objects as if they are ordinary objects.
But they will insist that this is an error. So sense-datum theories are not simply refuted as Harman seems to argue by pointing to the phenomenological fact that the objects of experience seem to be the ordinary things around us. And in perceptual experience we are not aware of ordinary objects but non-ordinary sense-data. Thus Awareness is false. Must a non-minimal sense-datum theorist deny Awareness? It seems not. Some sense-datum theorists introduce a distinction between direct and indirect awareness. The sense-datum theorist can say that we are indirectly aware of ordinary objects: that is, aware of them by in virtue of being aware of sense-data.
A sense-datum theorist who says this is known as an indirect realist or representative realist, or as someone who holds a representative theory of perception see Jackson , Lowe ; see also the entry on epistemological problems of perception. A theorist who denies that we are aware of mind-independent objects at all, directly or indirectly, but only of sense-data construed as mental entities, is known as a phenomenalist or an idealist about perception see Foster for a recent defence, see Crane and Farkas Section 2 for an introduction to the subject; and the entry on idealism.
On the face of it the indirect realist form of the sense-datum theory salvages something of our ordinary conception of perceptual experience, but securing Awareness. First, Openness is still being denied. Second, once we are given the distinction between direct and indirect perception, a defender of our ordinary conception of perceptual experience is likely to uphold Awareness in a more specific form, that is, as the idea that perceptual experience sometimes gives us direct awareness of ordinary objects. That is, the main theories of experience which uphold our ordinary conception of perceptual experience—intentionalism and naive realism—are both usually regarded as versions of direct realism.
The sense-datum theory was widely rejected in the second half of the 20 th century, though it still had its occasional champions in this period e. A number of objections have been made to the theory. For recent discussion see Silins A common objection in contemporary philosophy is to attack the Phenomenal Principle see Barnes —5 ; Anscombe The objection is that the Phenomenal Principle is fallacious. Rather, it is true because of specific phenomenological facts about perceptual experience.
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But this just means that theorists who reject the Phenomenal Principle are not disagreeing about whether the Phenomenal Principle involves a fallacy or about some semantic issue, but rather about the nature of experience itself. Another influential objection to sense-data comes from the prevailing naturalism of contemporary philosophy.
Naturalism or physicalism says that the world is entirely physical in its nature: everything there is supervenes on the physical, and is governed by physical law. Many sense-datum theorists are committed to the claim that non-ordinary sense-data are mind-dependent : objects whose existence depends on the existence of states of mind. Is this consistent with naturalism? If so, the challenge is to explain how an object can be brought into existence by the existence of an experience, and how this is supposed to be governed by physical law.
Many contemporary sense-datum theorists, however, will not be moved by this challenge, since they are happy to accept the rejection of naturalism as a consequence of their sense-data theory see Robinson , Foster On the other hand, one might think that there is no conflict here with naturalism, as long as experiences themselves are part of the natural order. But if sense-data are non-ordinary in being mind-independent but non-physical , then it is much less clear how naturalism can be maintained cf. Some philosophers agree with the Phenomenal Principle that whenever a sensory quality appears to be instantiated then it is instantiated, but deny that this entails the existence of sense-data.
Rather, they hold that we should think of these qualities as modifications of the experience itself Level 1. Hence when someone has an experience of something brown, something like brownness is instantiated, but in the experience itself, not an object. Part of the point of the adverbial theory, as defended by Ducasse and Chisholm was to do justice to the phenomenology of experience whilst avoiding the dubious metaphysical commitments the sense-datum theorists take on in responding to the Problem of Perception.
The only entities which the adverbialist needs to acknowledge are subjects of experience, experiences themselves, and ways these experiences are modified. This makes the theory appear less controversial than the sense-datum theory. In this broad sense, any conscious state of mind has qualia. This is the way the term is used in, e. Harman rejects mental paint, the idea of experience as involving mental paint is taken up and defended by Block It is relatively uncontroversial to say that there are qualia in the broad sense.
It can be misleading, however, to use the term in this way, since it can give rise to the illusion that the existence of qualia is a substantial philosophical thesis when in fact it is something which will be accepted by anyone who believes in phenomenal character. It is controversial to say that there are qualia in the narrow sense, though, and those who have asserted their existence have therefore provided arguments and thought-experiments to defend this assertion see Block , Peacocke Chapter 1 , Shoemaker The most natural way to understand this is that the experience is an event, and the modification of it is a property of that event.
Since this property is both intrinsic as opposed to relational or representational and phenomenal that is, consciously available then this way of understanding the adverbial theory is committed to the existence of qualia. An important objection to the adverbial theory has been proposed by Frank Jackson Consider someone who senses a brown square and a green triangle simultaneously. But how can it distinguish the state of mind it is describing in this way from that of sensing a brown triangle and a green square?
The characterisation fits that state of mind equally well. But how is the theory to do this without introducing objects of experience—the things which are brown and green respectively—or a visual field with a spatial structure? The challenge is whether the adverbial theory can properly account for the spatial structure and complexity in what is given in visual experience. See Tye for an attempt to respond to this challenge.
A related objection concerns the relationship between the adverbial theory and our ordinary conception of experience. The adverbial theorist might admit that, in a sense, we are aware of ordinary objects. And it is not as if such awareness is indirect or mediated by sense-data. However, it is not at all clear that the adverbialist is in a position to secure Openness.
For the adverbialist rejects not just the idea that experience has a genuine act-object structure, but the idea that the character of experience is even a presentation as of ordinary things and qualities. Qualities get into the picture, and are constitutive of phenomenal character, but not by being presented from outside of experience as qualities of things, as Openness would have it. How, then, can the adverbialist account even for the appearance of an act-object structure within experience, for Openness?
It is unclear how the adverbialist is to answer this question see Martin ; Crane And so it is unclear how much of an improvement the adverbial approach, and the qualia theory, is over the sense-datum theory. At Level 1, the intentional theory of experience treats perceptual experience as a form of intentionality conceived of as a form of mental representation hence it is also sometimes called the representationalist theory of experience.
An intentional mental state is normally understood, therefore, as one which is about, or represents, something in the world. At Level 2, this is put to work in explaining phenomenal character. Take an experience as of a churchyard covered in white snow. Why is this a case of things appearing white to one? It is not generally true that when a representation represents something as being F , there has to actually be something which is F.
Thus for the intentionalist, experience is representational in a way that contrasts with it being relational. Experience does not genuinely have an act-object structure. So intentionalism contrasts with the sense-datum theory: since it is not of the essence of experience or its character that it is relational, it is not of its essence that it is a relation to a non-ordinary sense-datum.
This is because these arguments hinge on the minimal sense-datum theory, or the Phenomenal Principle which the intentional theory is an alternative to. An illusory experience in which a white wall appears yellow to one is thus not conceived of as a case in which one is aware of a yellow sense-datum. It is instead conceived of as a case in which a white wall is represented as being yellow.
What about hallucinations? But what about the original form of the argument from hallucination? This argument does not rely on the Phenomenal Principle, yet its conclusion is that not even veridical experiences give us direct awareness of ordinary mind-independent objects. So how is the intentionalist to deal with this argument? The intentionalist accepts A but also B in a specific form. That is, they accept B where that is understood in terms of what Martin , has called the Common Kind Assumption, that is:.
CKA Whatever fundamental kind of mental event occurs when one veridically perceives, the very same kind of event could occur were one hallucinating. Take a veridical perception of a white snow covered churchyard for what it is.
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The experience involved in this perception, the intentionalist thinks, is fundamentally a matter of experientially representing the presence of a white snow covered churchyard. And this fundamental kind of mental event is exactly what is present in the subjectively indistinguishable hallucinatory case, for such hallucinatory experiences have the same representational nature as their veridical counterparts.
If so, the rejection of Awareness and our ordinary conception of perceptual experience is not far off. Since if we are not perceptually aware of ordinary objects in veridical experiences, it is unclear how any form of perceptual awareness of ordinary objects can be secured. However, the intentionalist can distinguish between two readings of the conclusion. On the first reading, we have. The intentionalist admits that not even veridical experiences are fundamentally cases of perceptual awareness of ordinary objects.
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Such experiences are fundamentally representational in a way that contrasts with them being relational. Veridical experiences may be the occasions for such awareness even if they are not themselves constituted by instances of such awareness. They can be occasions for such awareness precisely because they represent ordinary objects. In their very character they are about, directed on, the mind-independent world in contrast to both sense-datum theories, and adverbialist theories. We come to have direct perceptual awareness by having such experiences when the world also plays its part: when things in the world are as they are represented to be in the experience, and when the world is hooked up to the experience in an appropriate way.
This also helps us to see how even illusions can give us direct awareness of ordinary objects. Thus the intentionalist can respond to Problem of Perception: it has solutions to both arguments which animate the Problem. And the intentionalist response secures both aspects of our ordinary conception of perceptual experience. The intentionalist explains how experience satisfies Openness in terms of it having a certain sort of representational nature.
That is, a perceptual experience involves, in its very character, the presentation of ordinary mind-independent objects to a subject precisely because it is a matter of perceptual representation of ordinary aspects of the environment. And such aspects are represented as there , present. The character of experience is immediately responsive to the character of its objects because it is constituted, at least in part, by the way those objects are represented, at the time they are experienced.
Some of the most influential intentional theories are Anscombe , Armstrong , Pitcher , Peacocke , Harman , Tye , , Dretske , Lycan ; for more recent accounts, see Byrne , Siegel , Pautz and the entry on the contents of perception. Within analytic philosophy, the intentionalist theory of perception is a generalisation of an idea presented by G. Armstrong and George Pitcher Within the phenomenological tradition intentionality and perception had always been discussed together: see the entry on phenomenology.
Anscombe had drawn attention to the fact that perceptual verbs satisfy the tests for non-extensionality or intensionality see the entry on intensional transitive verbs. In neither case can we infer that there exists something Vladimir is thinking about, or that there is exists something he is experiencing.
This is the typical manifestation of intensionality. Anscombe regarded the error of sense-data and naive realist theories of perception as the failure to recognise this intensionality. Armstrong and Pitcher argued that perception is a form of belief. If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space.
But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought. If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events.
If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B.
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We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—. The following reflexions are based on the principle of relativity and on the principle of the constancy of the velocity of light. These two principles we define as follows:—. Let there be given a stationary rigid rod; and let its length be l as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod.
We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:—. Current kinematics tacitly assumes that the lengths determined by these two operations are precisely equal, or in other words, that a moving rigid body at the epoch t may in geometrical respects be perfectly represented by the same body at rest in a definite position. Let a ray of light depart from A at the time 4 , let it be reflected at B at the time , and reach A again at the time. Taking into consideration the principle of the constancy of the velocity of light we find that.
Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous. So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.
Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike.
Now to the origin of one of the two systems k let a constant velocity v be imparted in the direction of the increasing x of the other stationary system K , and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it; and that we thus obtain the co-ordinates x , y , z , and , , respectively. To any system of values x , y , z , t , which completely defines the place and time of an event in the stationary system, there belongs a system of values , , , , determining that event relatively to the system k , and our task is now to find the system of equations connecting these quantities.
In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.
We first define as a function of x ', y , z , and t. From the origin of system k let a ray be emitted at the time along the X-axis to x ', and at the time be reflected thence to the origin of the co-ordinates, arriving there at the time ; we then must have , or, by inserting the arguments of the function and applying the principle of the constancy of the velocity of light in the stationary system:—. It is to be noted that instead of the origin of the co-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x ', y , z.
An analogous consideration—applied to the axes of Y and Z—it being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity gives us. Since is a linear function, it follows from these equations that. With the help of this result we easily determine the quantities , , by expressing in equations that light as required by the principle of the constancy of the velocity of light, in combination with the principle of relativity is also propagated with velocity c when measured in the moving system. For a ray of light emitted at the time in the direction of the increasing.
But the ray moves relatively to the initial point of k , when measured in the stationary system, with the velocity c - v , so that. If we insert this value of t in the equation for , we obtain. In an analogous manner we find, by considering rays moving along the two other axes, that. If no assumption whatever be made as to the initial position of the moving system and as to the zero point of , an additive constant is to be placed on the right side of each of these equations. We now have to prove that any ray of light, measured in the moving system, is propagated with the velocity c , if, as we have assumed, this is the case in the stationary system; for we have not as yet furnished the proof that the principle of the constancy of the velocity of light is compatible with the principle of relativity.
At the time , when the origin of the co-ordinates is common to the two systems, let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K. If x , y , z be a point just attained by this wave, then. Transforming this equation with the aid of our equations of transformation we obtain after a simple calculation. The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system.
This shows that our two fundamental principles are compatible. In the equations of transformation which have been developed there enters an unknown function of v , which we will now determine. We call the co-ordinates, measured in the system , x ', y ', z ', and by a twofold application of our equations of transformation we obtain. Since the relations between x ', y ', z ' and x , y , z do not contain the time t , the systems K and are at rest with respect to one another, and it is clear that the transformation from K to must be the identical transformation.
We now inquire into the signification of. We give our attention to that part of the axis of Y of system k which lies between and. This part of the axis of Y is a rod moving perpendicularly to its axis with velocity v relatively to system K. Its ends possess in K the co-ordinates. The length of the rod measured in K is therefore ; and this gives us the meaning of the function.
From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if v and - v are interchanged. Hence follows that , or. It follows from this relation and the one previously found that , so that the transformation equations which have been found become. We envisage a rigid sphere 6 of radius R, at rest relatively to the moving system k , and with its centre at the origin of co-ordinates of k.
The equation of the surface of this sphere moving relatively to the system K with velocity v is.