Einstein for Everyone Lecture 5: E = mc 2 ; Philosophical Significance of Special Relativity

Transcription

1 Einstein for Everyone Lecture 5: E = mc 2 ; Philosophical Significance of Special Relativity Dr. Erik Curiel Munich Center For Mathematical Philosophy Ludwig-Maximilians-Universität

2 1 Dynamics Background Energy and Momentum 2 Historical Digression 3 Most Famous Equation 4 Philosophical Implications Nature of Physical Quantities Theory Change 5 Methodological Issues Verificationism Operationalism

3 Background Consequences of Relativity Relativity of Space and Time Relativity of Simultaneity Disagreement among observers regarding spatial distance and temporal intervals between events... what else is relative? Further Implications Dynamical descriptions of motion Disagreement regarding forces, masses, momenta Leads to E = mc 2

4 Background Introducing Dynamics Kinematics Description of motion in space and time, without regarding the causes of motion. (Anything goes!) Dynamics Account of the causes of motion, by use of dynamical laws. = Dynamics draws distinction between motion that is allowed according to the laws and motion that is not possible.

5 Energy and Momentum Force and Mass Key dynamical concepts of Newtonian physics: 1 Mass: property of an object that determines how hard it is to change its velocity. Greater mass more force needed to change the object s motion. 2 Momentum: measure of the quantity of motion, given by momentum = mass velocity 3 Force: roughly, measures strength of interaction between two bodies, rate of momentum exchange. Greater force exerted on a body greater change in its momentum.

6 Energy and Momentum Relativistic Dynamics? Key dynamical concepts of Einsteinian physics: 1 Mass 2 Momentum 3 Force Changes? - Relativistic correction factors pop up in definitions - Key principles relating the quantities remain true,... and they imply that E = mc 2

7 Energy and Momentum Changing Energy and Momentum Suppose a constant force is exerted on body A: how does this effect its momentum and energy? Momentum change = Force time Energy change = Force distance (Energy change usually called work ) Relativistic Version These principles still hold in relativity, but we have to take time dilation and length contraction into account...

8 Energy and Momentum Conservation Principles What is a conservation principle? System before interaction... Momentum i for Ball 1 + Momentum i for Ball 2 Total momentum i... and after Momentum f for Ball 1 + Momentum f for Ball 2 Total momentum f Conservation... of momentum: total momentum i = total momentum f... of energy: total energy i = total energy f

9 Energy and Momentum Conservation Principles Conservation of energy and momentum for closed systems Closed: isolated from environment, only exchange energy and momentum with other parts of the system Illustration : Center-of-Mass Theorem

10 1 Dynamics Background Energy and Momentum 2 Historical Digression 3 Most Famous Equation 4 Philosophical Implications Nature of Physical Quantities Theory Change 5 Methodological Issues Verificationism Operationalism

11 Lorentz (1895) Lorentz s Ether and Matter - Ether can set matter in motion (electromagnetic fields change motion of particles) - But matter has no effect on the immobile ether It is true that this conception would violate the principle of the equality of action and reaction because we do have grounds for saying that the ether exerts forces on ponderable matter but nothing, as far as I can see, forces us to elevate that principle to the rank of a fundamental law of unlimited validity. (Lorentz 1895) Momentum conservation doesn t apply to electromagnetism!

12 Poincaré s Puzzle Consider the following cases: 1 Gun firing a bullet 2 Flashlight emitting light beam Momentum conservation implies recoil in first case; what about second? Poincaré 1904 The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy.

13 Resolving the Puzzles Einstein s Response to Lorentz: There is no ether. But the electromagnetic field itself carries momentum. So the law of conservation of momentum applies to interactions between particles and electromagnetic fields! Einstein s Response to Poincaré: The apparatus will recoil, but Newton s principle does not fail to hold. Instead there is a mass associated with the projectile (via E = mc 2 )!

14 1 Dynamics Background Energy and Momentum 2 Historical Digression 3 Most Famous Equation 4 Philosophical Implications Nature of Physical Quantities Theory Change 5 Methodological Issues Verificationism Operationalism

15 Equivalence of Mass and Energy 1 Starting Question 1 Kinematics: relativistic velocity addition, c as an upper limit 2 Dynamics: change in momentum by adding force 2 Why doesn t constant force lead to unlimited momentum increase and unlimited velocity increase?

16 Unlimited Momentum Increase Norton s example: Ball bouncing elastically between two moving blocks Ball s velocity and momentum increases, able to accelerate past c!

17 Unlimited Momentum / Limited Velocity? Reconciliation? Momentum = mass velocity - Velocity increases towards c as upper limit - Momentum increases without limit -... so mass must increase without limit as well! Conclusion: adding energy to a body to change its motion (doing work on it) changes its mass

18 E = mc 2 Equivalence between mass and energy The mass of a body is a measure of its energy content (Einstein 1905) Add E units of energy = add E/c 2 units of mass

19 E = mc 2 Where does the c 2 term come from? Uniform force applied to body moving at c, increase in mass. Claim: per unit time, the energy E added to the body = mc 2 Energy = Force distance - Body moves nearly c in unit time - E = force c Combining: E = Force c = mc c = mc 2 Momentum = Force time - Momentum gained = mc - So mc = Force (for unit time)

20 From Kinematics to Dynamics What was once split asunder... - Spatial distance - (Absolute) time intervals... combined in new absolute quantity - Space-time interval Distinct dynamical concepts... - Mass - Energy - Momentum... combined into one quantity - Energy-momentum

21 From Kinematics to Dynamics Kinematics: Velocity Addition Einstein tax : incremental change of velocity smaller, closer to c For initial velocities < c, sum never reaches c Dynamics: Constant Force Force continues to push, contribute same amount of momentum Total momentum increases by same increments but velocity does not Mass changes as a function of velocity!

22 Momentum Conservation Imagine Al and Bob witness a collision: - Momentum conservation holds according to Al - Momentum conservation holds according to Bob But they disagree on velocities so there must be compensating disagreement on masses

23 Momentum Conservation Working through details (see Giulini 3.8): m 0 = rest mass m(v) = m 0 1 v2 c 2 (3.1)

24 Does the Inertia of a Body Depend on its Energy Content? What about internal rather than kinetic energy? Einstein (1905): E = mc 2 holds generally for all kinds of energy - Consider emission of light from a body - Have to associate energy and momentum to rest mass for conservation laws to hold (Giulini 3.9)

25 1 Dynamics Background Energy and Momentum 2 Historical Digression 3 Most Famous Equation 4 Philosophical Implications Nature of Physical Quantities Theory Change 5 Methodological Issues Verificationism Operationalism

26 Implications of Relativity Theory? 1 Methodological: how did Einstein do it? What should scientists do to make similar advances? 2 Philosophical: what does the new theory imply about views regarding space and time? change? Broader philosophical implications? 3 Historical: what does this case show about the development of ideas in physics? How do concepts change in physics?

27 Nature of Physical Quantities Relativity: What s In a Name? Theory of relativity everything is relative? - Fallacy of ambiguity: relative in Einstein s case refers to principle of relativity, not related to moral relativism (etc.) Theory of absolute invariants (Minkowski) - Everything is not relative in Einstein s theory! - Change in what is invariant, not elimination of invariant quantities

28 Theory Change Historical Implication: Pessimistic Meta-Induction 1687 Isaac Newton s Principia: laws of motion, gravity - Concepts of space, time, force, motion - Incredible empirical success for > 200 years 1905 Einstein, On the Electrodynamics of Moving Bodies - New concepts of space and time, Newton incorrect - Incredible empirical success for > 100 years another revolutionary change?

29 Theory Change Historical Implication: Pessimistic Meta-Induction Fate of past theories - Specify some standard for accepting a theory based on evidence. In the past, various scientific theories have met this standard (e.g. Newtonian gravity). - These theories are no longer accepted as true. Fate of current theories? - We currently have a number of theories that have met the standard for accepting a theory. - Why should we expect these to be true, given the fate of successful theories in the past?

30 Theory Change Historical Implication 1: Pessimistic Meta-Induction Pessimist - Successful theories of the past are no taken to be true - Why have confidence in currently successful theories? Optimistic response - Past theories still useful as approximations within a limited domain - New theories preserve successful prior theories

31 Theory Change Historical Implication 2: Conceptual Change... the physical referents of these Einsteinian concepts are by no means identical with those of the Newtonian concepts that bear the same name. (Newtonian mass is conserved; Einsteinian is convertible with energy. Only at low relative velocities may the two be measured in the same way, and even then they must not be conceived to be the same.) (Kuhn 1962, Structure of Scientific Revolutions, p. 102) To make the transition to Einstein s universe, the whole conceptual web whose strands are space, time, matter, force, and so on, had to be shifted and laid down again on nature whole. (Kuhn 1962, p. 149)

32 Theory Change Kuhn s Two-Process View 1 Normal Science Progress through detailed problem-solving Community accepts basic worldview, paradigm 2 Scientific Revolutions Fundamental, deep change in ideas, paradigm itself is in question No direct sense of cumulative progress

33 Theory Change Historical Implication 2: Conceptual Change Kuhn s Claims 1 Newtonian mass and Einsteinian mass different concepts, incommensurable 2 Holism: meaning of mass, space, time,... defined within entire theory 3 Newton to Einstein: example of Scientific Revolution Brief Replies - Approximate Newtonian ideas in the limit c - Careful analysis of tensions within Newtonian theory led to Einstein s discoveries

34 1 Dynamics Background Energy and Momentum 2 Historical Digression 3 Most Famous Equation 4 Philosophical Implications Nature of Physical Quantities Theory Change 5 Methodological Issues Verificationism Operationalism

35 Einstein s Method? How did Einstein make the discoveries in his 1905 paper? Is there a method that could be emulated in other domains?

36 Verificationism Verificationism Verifiability Criterion of Meaning The meaning of a statement consists in its means of verification. Verifiable vs. verified Versions of verifiability criterion defended by logical positivists (Carnap, Reichenbach), contemporaries of Einstein

37 Verificationism Verificationism Applied? Einstein contra Lorentz and others - Lorentz et al.: postulate ether with a definite rest frame... but generalized contraction effects make it inaccessible - Einstein: Ether state of rest superfluous unverifiable and therefore meaningless? Against verificationist reading - Einstein never directly says ether rest frame is meaningless

38 Operationalism Bridgman: Operationalism We must remain aware of these joints in our conceptual system if we hope to render unnecessary the services of the unborn Einsteins. (Bridgman, 1927) Concepts defined in terms of set of operations - Example: different concept of length for each different operation used to determine length Advantages of this approach - Fully specifying the operations helps to identify and eliminate false assumptions - Einstein s analysis of simultaneity

39 Operationalism Operationalism Assessed Concepts defined in terms of set of operations - Example: different concept of length for each different operation used to determine length Disadvantages of this approach - Proliferation of different concepts - Example: temperature as measured by mercury thermometer, thermocouple,... - But take these to be different measurements of same quantity

40 1 Dynamics Background Energy and Momentum 2 Historical Digression 3 Most Famous Equation 4 Philosophical Implications Nature of Physical Quantities Theory Change 5 Methodological Issues Verificationism Operationalism

41 Change and Becoming Views on the Nature of Time and Change Dynamic Views - Fundamental distinction between Past, Present and Future - Temporal Becoming: the present or the now advances Static Views - Distinction between Past, Present and Future merely perspectival - Block universe Our Question: Does relativity give us a reason to prefer the static view? Is there non-trivial becoming?

42 Change and Becoming Relativity and the Future Spaceship now is determinate with respect to Earth now (because the two events are simultaneous) Earth later is determinate with respect to spaceship now Assuming transitivity, Earth later is determinate with respect to Earth now

43 Change and Becoming The future is determined? Putnam (1967): argument in favor of static view - Argument regarding determinate generalizes - Every point in spacetime determinate with respect to every other point

44 I conclude that the problem of the reality and determinateness of future events is now solved. Moreover, it is solved by physics and not philosophy. We have learned that we live in a four-dimensional and not a three-dimensional world, and that space and time... are just two aspects of a single four-dimensional continuum.... Indeed, I do not believe that there are any longer any philosophical problems about Time; there is only the physical problem of determining the exact physical geometry of the four-dimensional continuum we inhabit. (Putnam 1967, 247)

45 Change and Becoming The future is determined? Status of Putnam s assumptions - Why does simultaneous with determinate? - Why accept transitivity? - Both based on introducing philosophical idea of determinateness, don t follow directly from relativity theory

46 Change and Becoming The future is determined? Alternative response - Relativity compatible with dynamic or static view Reconsidering the assumptions - Simultaneity relative to state of motion of observer - Determinateness relative to... X... : 1... state of motion of observer space-time point, the here and now...

47 Change and Becoming Stein 1968 The common fallacy of [Putnam s] arguments is their employment, in the context of the Einstein-Minkowski theory, of notions about time that are illegitimate in that theory.... The Einstein-Minkowski structure gives us (in a very clear mathematical sense) temporal relations, but no time simpliciter. In the context of special relativity, therefore, we cannot think of temporal evolution as the development of the world in time, but have to consider instead (as above) the more complicated structure constituted by, so to speak, the chronological perspective of each space-time point.

48 The leading principle that connects this mathematical structure with the notions of process and evolution (and justifies the use of our notion of becoming in relativistic space-time) is this: At a space-time point a there can be cognizance of or information or influence propagated from only such events as occur at points in the past of a. In the geometrical situation defined at the beginning of this section, the point c is not in the past of a (nor identical with a), and a is not in the past of c; therefore, no information or influence from or cognizance of either can occur at the other. This provides no warrant at all for the claim that the event at c must be real or determinate to an observer at a; and the fact that there is a time axis orthogonal to the direction from a to c (or a time-coordinate function having equal values at a and c) adds nothing. Change and Becoming Stein 1968, cont.

49 Change and Becoming Stein 1968, cont. The relation of b to c is of course just the same; the existence... of a time coordinate whose value is smaller at b than at c, does not put b in the absolute, or physical, or relativistically invariant sense in the past of c, because a time coordinate is not time. Neither a nor b is, in any physically significant sense, present (or past) for any observer at c regardless of his velocity for neither has already become for c (nor has c for them); but a has already become for b, and can influence it.

50 Causal Theory of Time Reichenbach on Time Hans Reichenbach ( ) The decisive argument in favor of defining time order in terms of causal order derives from Einstein s criticisms of simultaneity. It is well known that the Lorentz transformations... permit the reversal of time order of certain events, namely those which cannot be connected by causal chains. It follows that if time order were more than causal order... Einstein s relativity could not be accepted. (Reichenbach 1955, p. 25)

51 Causal Theory of Time Causal Theory of Time What is time? Event p is earlier than q =:... event p could causally affect q Advantages? Take causal connectability as basic, reductive definition of time Recover timelike / spacelike distinction crucial in relativity

52 Causal Theory of Time Assessment Norton s Response - Definition, reduction: illuminating if it reduces A to simpler concepts B, C,... - Is causation really more basic, simple than time? Other problems - Extension to Einstein s general theory of relativity