Manual Curved Space & the More Delicate Times

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She reflects on things around her, and the most eternal lessons come into view. Paul Conn, President, Lee University. For example, the duration between the point event of the clock displaying "2" and the event of the clock displaying "5" is 5 - 2 , namely 3. There are no units. The units eventually need to be specified. Any metric m for time, including alternatives to the m function above, is a two-place function from a pair of time coordinates to a real number, the so-called duration between the pair, such that, for any three point-instants with time coordinates a, b, and c, the function m obeys the following laws:.

If time's instants form a linear continuum, then between any two distinct instants, there are many others. So, if we have a digital clock that ticks every second, we know that between those ticks there are many other times not being displayed by the clock. Is one metric more natural than another?

Philosophers are interested in the underlying issue of whether the choice of a metric for a space is natural in the sense of being objective or whether its choice is always a matter of convention. We will now add more some detail to the above treatment of the metric for time and include a discussion of the interval for spacetime. It is very important in the following discussion to be sensitive to the difference between a physical space and a mathematical space. A mathematical 3-dimensional space about dollar sales of cell phones and names of salespersons and dates of sales also has a metric specifiying 'distances' or 'separations' or 'intervals' between points in that space.

A mathematical space is simply a collection of points, and a metric tells us the interval or "separation" between those points. Not every mathemtical space can have a metric, but for those that can, we have a clear idea of the conditions that must be satisfied by the metric. To have a metric space, we need there to be a function specifying the interval or "measure of separation" between pairs of points. The interval must obey certain precisely specified conditions. Suppose we want a metric for spatial distance on the surface of Earth.

We usually will not know in advance the distances between every two points on the globe, but we do know, in advance of measuring, that the distance between San Francisco and New York City will be less than or equal to the sum of the distance from San Francisco to Chicago plus the distance from Chicago to New York City. In our everyday Euclidean space, the interval between two points is the length of the straight line connecting them.

This interval is the spatial distance. However, the term "interval" is a technical term that does not always mean spatial distance. In spacetime, how is the interval different from spatial distance and temporal duration? The brief answer is that for a pair of point events happening in the same place, the interval is just the temporal duration between them, but for two point events happening at the same time, the interval is the spatial distance between them.

In general the interval contains information about space and information about time. Let's investigate what that interval is. Einstein's theory requires that every non-accelerating observer should agree on the same spacetime interval for every pair of point events. But let's elaborate on these remarks. In our actual spacetime, the theory of special relativity implies that two observers using reference frames that move relative to each other will correctly calculate different distances and different durations for pairs of point events, but they will calculate the same interval.

That is why the official name of the interval is the invariant relativistic interval. If we agree with physicists that what is objective about spacetime is what does not change with a change in the reference frame we use, then the interval is objective, but velocities, distances, and durations are not. The metric determines its geometry, and this geometry is intrinsic in the sense that it does not change as we change to another legitimate reference frame. If we select a standard clock and the standard metric for time, then we assume the duration between any two successive clock ticks is the same, provided the clock is stationary in the coordinate system where the clock readings are taken.

So, the duration between a pair of adjacent clicks yesterday is the same as between a pair tomorrow. A point of physical space is located by being assigned a coordinate. For doing quantitative science rather than merely qualitative science, we want the coordinate to be a number and not, say, a letter of the alphabet.

A coordinate for a point in two-dimensional space requires two numbers rather than just one; a coordinate for a point in n -dimensional space requires n independently assigned numbers, where n is a positive integer. You should prefer a real number rather than a rational number, even though no measuring tool could detect the difference, because, for example, a square one unit on a side has no real-valued diagonal, but does have a diagonal with a real-valued measure, namely the square root of two units. Without real values, the square won't have any length. Don't we want the diagonals of all our squares to have lengths?

06 - Curved Space - A Briefer History of Time - Stephen Hawking

Let's consider metrics in different dimensions. In a one-dimensional Euclidean space, namely for an ordinary straight line, the metric d for two points x and y is customarily given by. We have intuitions about a one-dimensional space for time coordinates. Note the application of the Pythagorean Theorem. If the space curves and so is not Euclidean, then a more sophisticated definition of the metric is required because we can no longer apply the Pythagorean Theorem, except perhaps in infinitesimal regions.

More generally, our intuitive idea of distance requires that, no matter how strange the space is, we want its metric function d to have the following distance-like properties. For any points p, q and r, the following five conditions must be satisfied:. Suppose you have a set of numbers representing all of a country's inter-city distances in the same units, say miles. The set of numbers will obey the above five conditions. If they don't, you have measured some distances incorrectly.

We generalize these intuitions about physical space to our mathematical spaces. Notice that there is no mention of the path the distance d is taken across; all the attention is on the point pairs themselves. Notice also that the distance from p to q is specified without mentioning how many points exist between p and q. Does your idea of distance imply that those conditions on d should be true?

So, does the 1D metric. This also satisfies our five conditions above. If space were to expand uniformly with time, then a cannot be a constant but must be a function of time, namely a t. After a metric is defined for a spacetime, the metric is commonly connected to empirical observations by saying that the readings taken with ideal meter sticks and ideal clocks and lasers and radar and so forth yield the values of the metric.

Now let's return to our discussion of the interval for spacetime. To have a metric for a 4-dimensional spacetime, we desire a definition of the interval between any two infinitesimally neighboring points in that spacetime. Less generally, consider an appropriate metric for the 4-D mathematical space that is used to represent the spacetime obeying the laws of special relativity theory.

It uses a Minkowski spacetime. What is an appropriate metric for this spacetime? Well, if we were just interested in the space portion of this spacetime, then the above 3D Euclidean metric is fine. But we've asked a delicate question because the fourth dimension of Minkowski's mathematical space is special, and it represents a time dimension and not a space dimension.

Because of time, our metric for spacetime needs to give what we have called the "interval" between any two point events, and not merely the spatial distance between the events. For any pair of point events at x',y',z',t' and x,y,z,t ,. If this is positive, we have a spacelike interval; when it is negative we have a timelike interval. Notice the plus and minus signs on the four terms. The minus on time indicates that the time dimension is not a spatial dimension. Here is another equally good candidate for the Lorentzian metric:. In this metric all the imaginary values of the interval are changed to real values, and vice versa.

Either metric is acceptable because all that is physically real and not merely a mathematical artifact in these mathematical formulas for metrics is ratios among the metric numbers. The interval is sensitive to both space and time. This is reflected in our comment earlier that, for any pair of point events happening in the same place, the interval is just the temporal duration between them, but for any two point events happening at the same time, the interval is the spatial distance between them.

That is why, distance in a Minkowski diagram usually does not correspond directly to distance in the space that obeys special relativity. Finally, any correct clock measures the interval along its spacetime trajectory. Two synchronized clocks will give the same reading if they are both stationary, but otherwise not.

And the difference in their readings is said to be due to time dilation. The interval of spacetime between two point events is complicated because its square can be negative. In ordinary space, if the space interval between two events is zero, then the two events happened at the same time and place, but in spacetime, if the spacetime interval between two events is zero, this means only that there could be a light ray connecting them.

It is because the spacetime interval between two events can be zero even when the events are far apart in distance that the term "interval" is very unlike what we normally mean by the term "distance. All the events that have a zero spacetime interval from some event e constitute e 's two light cones. If event 2 is outside the light cones of event 1, then event 2 is said to occur in the " absolute elsewhere" of event 1.

In that case, neither event could have affected the other by a causal influence traveling less than the speed of light. And, you as the analyst are free to choose a coordinate system in which event 1 happens first, or another coordinate system in which event 2 happens first, or even a coordinate system in which the two are simultaneous. But once the coordinate system is chosen, then this choice fixes the happens-before relation for all point-events.

The ticking marks off congruent, invariant intervals. If the clock is stationary in its own inertial reference frame, then x' - x is zero, and so are y' - y and z' - z'; so, the clock measures the quantity t' - t.

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What if we turn now from special relativity to general relativity? Adding space and time dependence particularly the values of mass-energy and momentum at points to each term of the Lorentzian metric produces the metric for general relativity. That metric requires more complex tensor equations; these put a multiplication factor g in front of each of the products of the differential displacements such as x' - x 2 and x' - x y' - y , and the mathematical difficulty of the description escalates.

For a helpful presentation of the details of the interval and the metric, see Maudlin , especially chapter 4. In , the mathematician Hermann Minkowski remarked that "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Broad countered that the discovery of spacetime did not break down the distinction between time and space but only their independence or isolation.

He argued that their lack of independence does not imply a lack of reality. Nevertheless, there is a deep sense in which time and space are "mixed up" or linked. This is evident from the Lorentz transformations of special relativity that connect the time t in one inertial frame with the time t' in another frame that is moving in the x direction at a constant speed v. In this Lorentz equation, t' is dependent upon the space coordinate x and the speed. In this way, time is not independent of either space or speed.

Each frame has its own way of splitting up spacetime into its space part and its time part. The reason why time is not partly space is that, within a single frame, time is always distinct from space. Time is a distinguished dimension of spacetime, not an arbitrary dimension. What being distinguished amounts to, speaking informally, is that when you set up a rectangular coordinate system on spacetime with an origin at, say, some important event, you may point the x-axis east or north or up, but you may not point it forward in time—you may do that only with the t-axis, the time axis.

Yes and no; it depends on what you are talking about. Time is the fourth dimension of 4-d spacetime, but time is not the fourth dimension of physical space because that space has only three dimensions. In 4-d spacetime, the time dimension is special and unlike any of the other three dimensions because the time dimension has a direction and a space dimension does not.

Mathematicians have a broader notion of the term "space" than the average person; and in their sense a space need not consist of places, that is, geographical locations. Not paying attention to the two meanings of the term "space" is the source of all the confusion about whether time is the fourth dimension. Newton treated space as three dimensional, and treated time as a separate one-dimensional space. He could have used Minkowski's idea, if he had thought of it, namely the idea of treating spacetime as four dimensional.

The mathematical space used by mathematical physicists to represent physical spacetime that obeys the laws of relativity is four dimensional; and in that mathematical space, the space of places is a 3D sub-space and time is another 1D sub-space. Minkowski was the first person to construct such a mathematical space, although in H. Wells treated time informally as a fourth dimension in his novel The Time Machine. In any coordinate system on spacetime, mathematicians of the early twentieth century believed it was necessary to treat a point event with at least four independent numbers in order to account for the four dimensionality of spacetime.

Actually this appeal to the 19th century definition of dimensionality, which is due to Bernhard Riemann, is not quite adequate because mathematicians have subsequently discovered how to assign each point on the plane to a point on the line without any two points on the plane being assigned to the same point on the line. The idea comes from the work of Georg Cantor. Because of this one-to-one correspondence, the points on a plane could be specified with just one number.

If so, then the line and plane must have the same dimensions according to the Riemann definition of dimension. To avoid this problem and to keep the plane being a 2D object, the notion of dimensionality of a space has been given a new, but rather complex, definition. Every reference frame has its own physical time, but the question is intended in another sense.

At present, physicists measure time electromagnetically. They define a standard atomic clock using periodic electromagnetic processes in atoms, then use electromagnetic signals light to synchronize clocks that are far from the standard clock. In doing this, are physicists measuring '"electromagnetic time" but not other kinds of physical time? In the s, the physicists Arthur Milne and Paul Dirac worried about this question. Independently, they suggested there may be very many time scales. For example, there could be the time of atomic processes and perhaps also a time of gravitation and large-scale physical processes.

Clocks for the two processes might drift out of synchrony after being initially synchronized, yet there would be no reasonable explanation for why they don't stay in synchrony. Ditto for clocks based on the pendulum, on superconducting resonators, on the spread of electromagnetic radiation through space, and on other physical principles.

Just imagine the difficulty for physicists if they had to work with electromagnetic time, gravitational time, nuclear time, neutrino time, and so forth. Current physics, however, has found no reason to assume there is more than one kind of time for physical processes. In , physicists did reject the astronomical standard for the atomic standard because the deviation between known atomic and gravitation periodic processes such as the Earth's rotations and revolutions could be explained better assuming that the atomic processes were the most regular of these phenomena.

But this is not a cause for worry about two times drifting apart. Physicists still have no reason to believe a gravitational periodic process that is not affected by friction or impacts or other forces would ever drift out of synchrony with an atomic process such as the oscillations of a quartz crystal, yet this is the possibility that worried Milne and Dirac. Physical time is not relative to any observer's state of mind.

Wishing time will pass does not affect the rate at which the observed clock ticks. On the other hand, physical time is relative to the observer's reference system —in trivial ways and in a deep way discovered by Albert Einstein. In a trivial way, time is relative to the chosen coordinate system on the reference frame. For example, it depends on the units chosen as when the duration of some event is 34 seconds if seconds are defined to be a certain number of ticks of the standard clock , but is 24 seconds if seconds are defined to be a different number of ticks of that standard clock.

Similarly, the difference between the Christian calendar and the Jewish calendar for the date of some event is due to a different unit and origin. Also trivially, time depends on the coordinate system when a change is made from Eastern Standard Time to Pacific Standard Time. These dependencies are taken into account by scientists but usually never mentioned.

For example, if a pendulum's approximately one-second swing is measured in a physics laboratory during the autumn night when the society changes from Daylight Savings Time back to Standard Time, the scientists do not note that one unusual swing of the pendulum that evening took a negative fifty-nine minutes and fifty-nine seconds instead of the usual one second.

Isn't time relative to the observer's coordinate system in the sense that in some reference frames there could be fifty-nine seconds in a minute? No, due to scientific convention, it is absolutely certain that there are sixty seconds in any minute in any reference frame.

How long an event lasts is relative to the reference frame used to measure the time elapsed, but in any reference frame there are exactly sixty seconds in a minute because this is true by definition. Similarly, you do not need to worry that in some reference frame there might be two gallons in a quart. In a deeper sense, time is relative, not just to the coordinate system, but to the reference frame itself. That is Einstein's principal original idea about time. Einstein's special theory of relativity requires physical laws not change if we change from one inertial reference frame to another.

In technical-speak Einstein is requiring that the statements of physical laws must be Lorentz-invariant. The equations of light and electricity and magnetism Maxwell electrodynamics are Lorentz-invariant, but those of Newton's mechanics are not, and Einstein eventually figured out that what needs changing in the laws of mechanics is that temporal durations and spatial intervals between two events must be allowed to be relative to which reference frame is being used. There is no frame-independent duration for an event extended in time.

To be redundant, Einstein's idea is that without reference to the frame, there is no fixed time interval between two events, no 'actual' duration between them. This idea was philosophically shocking as well as scientifically revolutionary. Einstein illustrated his idea using two observers, one on a moving train in the middle of the train, and a second observer standing on the embankment next to the train tracks.

If the observer sitting in the middle of the rapidly moving train receives signals simultaneously from lightning flashes at the front and back of the train, then in his reference frame the two lightning strikes were simultaneous. But the strikes were not simultaneous in a frame fixed to an observer on the ground. This outside observer will say that the flash from the back had farther to travel because the observer on the train was moving away from the flash. If one flash had farther to travel, then it must have left before the other one, assuming that both flashes moved at the same speed.

Therefore, the lightning struck the back of the train before the lightning struck the front of the train in the reference frame fixed to the tracks. Let's assume that a number of observers are moving with various constant speeds in various directions. Consider the inertial frame of reference in which each observer is at rest in his or her own frame.

Which of these observers will agree on their time measurements? Only observers with zero relative speed will agree. Observers with different relative speeds will not, even if they agree on how to define the second and agree on some event occurring at time zero the origin of the time axis. If two observers are moving relative to each other, but each makes judgments from a reference frame fixed to themselves, then the assigned times to the event will disagree more, the faster their relative speed. All observers will be observing the same objective reality, the same event in the same spacetime, but their different frames of reference will require disagreement about how spacetime divides up into its space part and its time part.

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This relativity of time to reference frame implies that there be no such thing as The Past in the sense of a past independent of reference frame. This is because a past event in one reference frame might not be past in another reference frame. However, this frame relativity usually isn't very important except when high speeds or high gravitational field strengths are involved.

In some reference frame, was Adolf Hitler born before George Washington? No, because the two events are causally connectible. That is, one event could in principle have affected the other since light would have had time to travel from one to the other. We can select a reference frame to reverse the usual Earth-based order of two events only if they are not causally connectible, that is, only if one event is in the absolute elsewhere of the other. Despite the relativity of time to a reference frame, any two observers in any two reference frames should agree about which of any two causally connectible events happened first.

The relativity of simultaneity is the feature of spacetime in which two different reference frames moving relative to each other will disagree on which events are simultaneous. This implies simultaneity is not an objectively real relationship among all the events. How do we tell the time of occurrence of an event that is very far away from us? We assign the time when something is occurring far away from us by subtracting, from the time we noticed it, the time it took the signal to travel all that way to us. For example, we see a flash of light at time t arriving from a distant place P.

When did the flash occur back at P? Let's call that time t p. Here is how to compute t p. Suppose we know the distance x from us to P. Then the flash occurred at t minus the travel time for the light. In this way, we know what events on the distant Sun are simultaneous with what clicks on our Earth clock.

The deeper problem is that other observers will not agree with us that the event on the Sun occurred when we say it did. The diagram below illustrates the problem. Let's assume that our spacetime obeys the special theory of relativity. There are two light flashes that occur simultaneously, with Einstein at rest midway between them in this diagram. The Minkowski diagram represents Einstein sitting still in the reference frame marked by the coordinate system with the thick black axes while Lorentz is not sitting still but is traveling rapidly away from him and toward the source of flash 2. Because Lorentz's timeline is a straight line, we can tell that he is moving at a constant speed.

The two flashes of light arrive at Einstein's location simultaneously, creating spacetime event B. However, Lorentz sees flash 2 before flash 1. That is, the event A of Lorentz seeing flash 2 occurs before event C of Lorentz seeing flash 1. So, Einstein will readily say the flashes are simultaneous, but Lorentz will have to do some computing to figure out that the flashes are simultaneous in the Einstein frame because they won't "look" simultaneous to Lorentz. However, if we'd chosen a different reference frame from the one above, one in which Lorentz is not moving but Einstein is, then Lorentz would be correct to say flash 2 occurs before flash 1 in that new frame.

So, whether the flashes are or are not simultaneous depends on which reference frame is used in making the judgment. The relativity of simultaneity is philosophically less controversial than the conventionality of simultaneity. To appreciate the difference, consider what is involved in making a determination regarding simultaneity. Given two events that happen essentially at the same place, physicists assume they can tell by direct observation whether the events happened simultaneously, assuming they are in a space obeying special relativity. If we don't see one of them happening first, then we say they happened simultaneously, and we assign them the same time coordinate.

The determination of simultaneity is more difficult if the two happen at separate places, especially if they are very far apart. One way to measure operationally define simultaneity at a distance is to say that two events are simultaneous in a reference frame if unobstructed light signals from the two events would reach us simultaneously when we are midway between the two places where they occur, as judged in that frame.

This is the operational definition of simultaneity used by Einstein in his theory of special relativity. The "midway" method described above has a significant presumption: This presupposition is about the conventionality, rather than relativity, of simultaneity. To pursue the issue of which event here is simultaneous with which event there, suppose the two original events are in each other's absolute elsewhere; that is, they could not have affected each other.

Einstein noticed that there is no physical basis for judging the simultaneity or lack of simultaneity between these two events, and for that reason he said we rely on a convention when we define distant simultaneity as we do.


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Hilary Putnam, Michael Friedman, and Graham Nerlich object to calling it a convention—on the grounds that to make any other assumption about light's speed would unnecessarily complicate our description of nature, and we often make choices about how nature is on the basis of simplification of our description. They would say there is less conventionality in the choice than Einstein supposed. The "midway" method is not the only way to define simultaneity.

Consider a second method, the "mirror reflection" method. Select an Earth-based frame of reference, and send a flash of light from Earth to Mars where it hits a mirror and is reflected back to its source.


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  • The flash occurred at The light traveled the same empty, undisturbed path coming and going. At what time did the light flash hit the mirror? The answer involves the so-called conventionality of simultaneity. All physicists agree one should say the reflection event occurred at The controversial philosophical question is whether this is really a convention.

    Einstein pointed out that there would be no inconsistency in our saying that it hit the mirror at Let's explore the reflection method that is used to synchronize a distant, stationary clock so that it reads the same time as our clock. Let's draw a Minkowski diagram of the situation and consider just one spatial dimension in which we are at location A with the standard clock for the reference frame. The distant clock we want to synchronize is at location B. See the following diagram. The fact that the timeline of the B-clock is parallel to the time axis shows that the clock there is stationary.

    We will send light signals in order to synchronize the two clocks. Send a light signal from A at time t 1 to B, where it is reflected back to us, arriving at time t 3. Then the reading t r on the distant clock at the time of the reflection event should be t 2 , where. He said this assumption is a convention, the so-called conventionality of simultaneity , and isn't something we could check to see whether it is correct.

    Either way, the average travel speed to and from would be c. Notice how we would check whether the two light speeds really are the same. We would send a light signal from A to B, and see if the travel time was the same as when we sent it from B to A. But to trust these times we would already need to have synchronized the clocks at A and B.

    However, some researchers suggest that there is a way to check on the light speeds and not simply presume they are the same. Transport one of the clocks to B at an infinitesimal speed. Going this slow, the clock will arrive at B without having its proper time deviate from that of the A-clock. That is, the two clocks will be synchronized even though they are distant from each other.

    Now the two clocks can be used to find the time when a light signal left A and the time when it arrived at B. The time difference can be used to compute the light speed. This speed can be compared with the speed computed for a signal that left B and then arrived at A. For more discussion of this controversial issue of conventionality in relativity, see pp.

    What does it mean to say the human condition is one in which you never will be able to affect an event outside your forward light cone? With any action you take, the speed of transmission of your action to its effect cannot move faster than c. It is the maximum speed in any reference frame. It is the speed of light and the speed of anything else with zero rest mass; it is also the speed of any electron or quark at the big bang before the Higgs field appeared and slowed them down.

    Here is a visual representation of the human condition according to the special theory of relativity, whose spacetime can always be represented by a Minkowski diagram of the following sort:. The absolutely past events the green events in the diagram above are the events in or on the backward light cone of your present event, your here-and-now.

    The backward light cone of event Q is the imaginary cone-shaped surface of spacetime points formed by the paths of all light rays reaching Q from the past. The events in your absolute past are those that could have directly or indirectly affected you, the observer, at the present moment. The events in your absolute future are those that you could directly or indirectly affect. An event's being in another event's absolute past is a feature of spacetime itself because the event is in the point's past in all possible reference frames.

    The feature is frame-independent. For any event in your absolute past, every observer in the universe who isn't making an error will agree the event happened in your past. Not so for events that are in your past but not in your absolute past. Past events not in your absolute past will be in what Eddington called your "absolute elsewhere. Your absolute elsewhere is the region of spacetime that is neither in nor on either your forward or backward light cones.

    No event here now, can affect any event in your absolute elsewhere; and no event in your absolute elsewhere can affect you here and now. A single point's absolute elsewhere, absolute future, and absolute past partition all of spacetime beyond the point into three disjoint regions. If point A is in point B's absolute elsewhere, the two events are said to be "spacelike related. For any two events in spacetime, one can tell in principle whether they are time-like, space-like, or light-like separated, and this is an objective feature of the pair that won't change with a change in reference frame.

    So denominating one lobe at p the future light-cone and the other the past light-cone settles also the distinction between the future and past directions at all other points of space-time. Such a space-time is called temporally orientable. The past light cone looks like a cone in small regions. However, the past light cone is not cone-shaped at the cosmological level but has a pear-shape because all very ancient light lines must have originated from the infinitesimal volume at the big bang. Time dilation is about two synchronized clocks getting out of synchrony due either to their relative motion or due to their being in regions of different gravitational field strengths.

    Time dilation due to difference in speeds is described by Einstein's special theory of relativity. Time dilation due to difference in acceleration or difference in travel through a varying gravitational field is described by Einstein's general theory of relativity. This section focuses on just the time dilation described by special relativity, namely time dilation due to speed. According to special relativity, two properly functioning, stationary clocks once properly synchronized will stay in synchrony no matter how far away from each other they are.

    But if one clock moves and the other does not, then the moving clock will tick slower than the stationary clock, as measured in the inertial reference frame of the stationary clock. This slowing due to motion is called "time dilation.

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    Your mass also increases by this amount. In addition, you are 7 times thinner along your direction of motion than you were when stationary. If you move at Suppose your twin sister's spaceship travels to and from a star one light year away while you remain on Earth. It takes light from your Earth-based flashlight two years to go there, reflect in a mirror and arrive back on Earth, so your clock will say your sister's trip took longer than two years or longer than the travel time for light.

    But if the spaceship is fast, she can make the trip in less than two years, according to her own clock. We sometimes speak of time dilation by saying time itself is "slower. According to special relativity, time dilation due to motion is relative in the sense that if your spaceship moves at constant speed past mine so fast that I measure your clock to be running at half speed, then you will measure my clock to be running at half speed also.

    This is not contradictory because we are making our measurements in different inertial frames. If one of us is affected by different gravitational field strengths or undergoes acceleration, then that person is not in an inertial frame and the results are slightly different, and general relativity is needed to describe what happens. Both these types of time dilation play a significant role in time-sensitive satellite navigation systems such as the Global Positioning System. The atomic clocks on the satellites must be programmed to compensate for the relativistic dilation effects of both gravity and motion.

    According to special relativity, if you are in the center of the field of a large sports stadium filled with spectators, and you suddenly race to the exit door at constant, high speed, everyone else in the stadium will get thinner in the frame fixed to you now than they were originally in the frame fixed to you before you left for the exit. The picture describes the same wheel in different colors: According to the general theory of relativity, gravitational differences affect time by dilating it.

    Observers in a less intense gravitational potential find that clocks in a more intense gravitational potential run slow relative to their own clocks. People in ground floor apartments outlive their twins in penthouses, all other things being equal. Basement flashlights will be shifted toward the red end of the visible spectrum compared to the flashlights in attics. This effect is known as the gravitational red shift.

    Informally, one speaks of gravity bending light rays around massive objects, but more accurately it is the spacetime that bends, and as a consequence the light's path is bent, too. The light simply follows the shortest path through space, a path called a "geodesic," and when spacetime curves the shortest paths are no longer Euclidean straight lines.

    Curved Space & the More Delicate Times by James P. Kain - Paperback | Souq - UAE

    There is a brief way to make these same points. Spacetime in the presence of matter or energy is curved. So, time is curved, too, and we see this curvature as time dilation effects. A black hole is highly compressed matter-energy whose gravitational field strength is strong enough that it warps spacetime so severely that all objects falling into the hole cannot get back out. Even light from inside cannot escape according to relativity theory, but not quantum theory once it is inside this point of no return. The point of no return is called the "event horizon. When that object reaches the horizon, its personal time completely stops, as judged by external observers.

    Space's warp around the Earth is much less powerful than the warp at a black hole, and so the Earth has no event horizon or point of no return for infalling material. Gravitational time dilation occurs around any massive object, but it is most extreme at the surface of the black hole. If you were in a spaceship near a black hole but outside its event horizon, your time warp would be more severe the longer you stayed in the vicinity and also the closer you got to the event horizon. If you fell into the black hole, then external observers would say your time stopped at the horizon, but from your perspective, you'd fall right through the horizon and continue.

    Relativity theory implies the black hole's center has a singularity, but quantum theory disallows this. If a black hole is turning or twisting, as most are, then inside the event horizon there inevitably will be objects whose worldlines are closed time-like curves, and so these objects undergo past time travel, although they cannot escape the hole. Perhaps an even odder temporal feature is that it is better to think of a person or object, once it arrives inside the black hole's event horizon, as aging toward the center rather than as falling toward it.

    This is because inside the horizon the roles of time and space are switched. The center is not a place in space; it is a moment when time ends. Trying to avoid the center by switching on your rockets will be as pointless as getting in a car here on Earth and driving fast in order to avoid tomorrow afternoon. For more on why time and space switch roles inside the black hole, see https: There are equally startling visual effects. A light ray can circle a black hole once or many times depending upon its angle of incidence. A light ray grazing a black hole can leave at any angle, so a person viewing a black hole from outside can see multiple copies of the rest of the universe at various angles.

    Quantum theory applied to black holes implies they are not totally black because whatever material does fall in eventually does get back out, though in a radically transformed manner. The infalling matter very slowly leaks back out as random Hawking radiation, and so eventually the hole will evaporate and disappear unless new material keeps falling in. The bigger the hole, the slower it evaporates; any hole more massive than Mt.

    Everest will take longer than eighteen billion years to evaporate even if nothing new falls in. The information of any infalling material remains on the hole's surface and will be re-emitted as the hole evaporates. As the size of the hole gets smaller its Hawking Radiation rises in frequency. When the hole is very tiny, about the size of the wavelength of light, its Hawking becomes white, producing a white black hole. The paradox is an argument about time dilation that uses the theory of relativity to produce an apparent contradiction.

    Consider two twins at rest on Earth with their clocks synchronized. One twin climbs into a spaceship and flies far away at a high, constant speed, then reverses course and flies back at the same speed. An application of the equations of relativity theory implies that because of time dilation the twin on the spaceship will return and be younger than the Earth-based twin. The argument for the twin paradox is that it is all relative. That is, either twin could regard the other as the traveler. If so, then when the twins reunite, each will be younger than the other.

    In brief, the solution to the paradox is that the two situations are not sufficiently similar, and that in both situations, the twin who stays home outside of the spaceship maximizes his or her proper time and so is always the older twin when the two reunite. This solution to the paradox has nothing to do with a proper or improper choice of reference frames. Let's look at more details of this solution. Herbert Dingle famously argued in the s that this paradox reveals an inconsistency in special relativity.

    Almost all philosophers and scientists now agree that it is not a true paradox, in the sense of revealing an inconsistency within relativity theory, but is merely a complex puzzle that can be adequately solved within relativity theory. There have been a variety of suggestions in the relativity textbooks on how to understand the paradox. Here is one, diagrammed below. The principal suggestion for solving the paradox is to note that there must be a difference in the proper time taken by the twins because their behavior is different, as shown in their two worldlines above.

    The coordinate time, that is, the time shown by clocks fixed in space in the coordinate system, is the same for both travelers. Their proper times are not the same. The longer line in the graph represents a longer path in space but a shorter duration of proper time. The length of the line representing the traveler's path in spacetime in the above diagram is not a correct measure of the traveler's proper time. Instead, the number of dots in the line is a measure of the proper time for the traveler.

    The spacing of the dots represents a tick of a clock in that worldline and thus represents the proper time elapsed along the worldline. The diagram shows how sitting still on Earth is a way of maximizing the proper during the trip, and it shows how flying near light speed in a spaceship away from Earth and then back again is a way of minimizing the proper time, even though if you paid attention only to the shape of the worldlines in the diagram and not to the dot spacing within them you might think just the reverse.

    This odd feature of the geometry is why Minkowski geometry is not Euclidean. So, the conclusion of the analysis of the paradox is that its reasoning makes the mistake of supposing that the situation of the two twins is the same as far as elapsed proper time is concerned. One twin's spacetime trajectory is longer, and that is the twin who is younger upon reunion. As our particular example above is set up, only one twin feels the acceleration at the turnaround point, but this is irrelevant to the solution to the paradox, despite what such famous physicists as Richard Feynman have said. The paradox could be restated in a form that requires both twins to feel the same acceleration.

    Also, the force felt by the spaceship twin is not what "forces" that twin to be younger. Nothing is forcing the twin to be younger other than the length of the twin's worldline. For more discussion of the paradox, and the criticism of Feynman, see Maudlin , pp. See the left diagram of his Fig. This question is asking how we coordinatize the four-dimensional manifold. The manifold is a collection of points technically, a topological space which behaves as a Euclidean space in neighborhoods around any point. Coordinates are assigned to points. Points cannot be added, subtracted, or squared, but their coordinates can be.

    Coordinates applied to the space are not physically real; they are tools used by the analyst, the physicist. In other words, they are invented, not discovered. Isaac Newton conceived of points of space and time as absolute in the sense that they retained their identity over time. Modern physics does not have that conception of points; points are identified relative to objects, for example the halfway point between this object and that object.

    In the late 16th century, the Italian mathematician Rafael Bombelli interpreted real numbers as lengths on a line and interpreted addition, subtraction, multiplication, and division as movements along the line. This work eventually led to our assigning real numbers to both instants and durations.

    Every event needs to be assigned a time coordinate. To justify the assignment of time numbers called time coordinates or dates or clock readings to instants, we cannot literally paste a number to an instant. What we do instead is rather complicated. For some of the details, the reader is referred to Maudlin , pp. Every event on the world-line of the master clock will be assigned a t -coordinate by the clock. Extending the t -coordinate to events off the trajectory of the master clock requires making use of Intuitively, two clocks are co-moving if they are both on inertial trajectories and are neither approaching each other nor receding from each other.

    An observer situated at the master clock can identify a co-moving inertial clock by radar ranging. That is, the observer sends out light rays from the master clock and then notes how long it takes according to the master clock for the light rays to be reflected off the target clock and return. If the target clock is co-moving, the round-trip time for the light will always be the same. Co-moving inertial clocks do not generally exist, according to general relativity, so the issue of assignments of time coordinates becomes quite complicated in the real world.

    This article will highlight only a few aspects of the assignment process. The assignment process assumes that the structure of the set of instantaneous events is the same as, or embeddable within, the structure of our time numbers. The structure of our time numbers is the structure of real numbers. Showing that this is so is called "solving the representation problem" for our theory of time measurement.

    This article won't go into detail on how to solve this problem, but the main idea is that the assignment of coordinates should reflect the structure of the space, namely its geometrical structure, which includes its topological structure, differential structure, affine structure, and metrical structure. For example, to measure any space, including a one-dimensional space of time, we need a metrification for the space.

    The metrification assigns location coordinates to all points and assigns distances between all pairs of points, when units are added. A metrification for time assigns dates to the points we call point-instants of time; these assignments are called time coordinates. Normally we use a clock to do this. Point instants get assigned a unique real number coordinate, and the metric or duration between any two of those point instants is found by subtracting their time coordinates from each other. The duration is the absolute value of the numerical difference of their coordinates, that is t B - t A where t B is the time coordinate of event B and t A is the time coordinate of A, for any pair of events A and B once units are added to the numbers.

    For ease of application of calculus to physical change, it is very important that nearby points get assigned nearby numbers so that all the coordinates change continuously as the point changes continuously in the space. Let's reconsider the question of metrification in more detail, starting with the assignment of locations to points.

    Any space is a collection of points. In a space that is supposed to be time, these points are the instants, and the space for time is presumably to be linear locally. Before discussing time coordinates specifically, let's consider what is meant by assigning coordinates to a mathematical space, one that might represent either physical space, or physical time, or spacetime, or the two-dimensional mathematical space in which we graph the relationship between the price of rice in China and the wholesale price of tulips in Holland. In a one-dimensional space, such as a curving line, we assign unique coordinate numbers to points along the line, and we make sure that no point fails to have a coordinate.

    For a two-dimensional space, we assign ordered pairs of real numbers to points. For a 3D space, we assign ordered triplets of numbers. Why numbers and not letters? If we assign letters instead of numbers, we can not use the tools of mathematics to describe the space. But even if we do assign numbers, we cannot assign any coordinate numbers we please.

    If the space has a certain geometry, then we have to assign numbers that reflect this geometry. If event A occurs before event B, then the time coordinate of event A, namely t A , must be less than t B. Here is the fundamental method of this analytic geometry:. Consider a space as a class of fundamental entities: The class of points has "structure" imposed upon it, constituting it as a geometry—say the full structure of space as described by Euclidean geometry. We can then study the geometry by studying, instead, the structure of the new associated system [of coordinates].