Egan, Greg - Foundations 2 - From Special To General

VIP免费

2024-12-23 0 0 139.83KB 21 页 5.9玖币

侵权投诉

Foundations

by Greg Egan

2: From Special to General

The first article in this series described some of the ways in which the geometry of

spacetime affects travellers moving (relative to their destinations, or each other) at a

substantial fraction of the speed of light. By generalising from the Euclidean metric,

which captures such familiar aspects of geometry as Pythagoras's Theorem, to the

Minkowskian metric suggested by the fact that the speed of light in a vacuum is the same

for everyone, we analysed the “rotated” view of spacetime that two observers in relative

motion have with respect to each other, and derived formulas for time dilation, Doppler

shift and aberration.

This article and the next will build the framework needed to provide a similar

account of the strange effects that have been predicted to take place in the vicinity of a

black hole. To do this, we need to generalise yet again: from flat geometry, to curved.

Gravity as Spacetime Curvature

The basic premise of general relativity is simple: the correct way to account for the

acceleration of objects due to gravity is to consider spacetime to be curved in the presence

of matter and energy. How does curvature explain acceleration? If two explorers set off

from different points on the Earth's equator, and both head north, their paths will grow

steadily closer together, despite the fact that they started out in the same direction. In

spacetime, if two nearby stars start out being motionless with respect to each other, their

world lines will draw closer together, despite the fact that those world lines were initially

pointing in the same direction. We could say that the force of gravity is pulling the stars

together … but we don't say there's a “force” acting on the explorers, do we? Of course,

the two-dimensional surface of the Earth is a visibly curved object embedded in a larger

(and more or less flat) space, but we have no reason to believe that spacetime is

embedded in anything larger. Rather, general relativity assumes that whatever gives rise

to spacetime geometry in the first place is tied up with the presence of matter and energy

in such a way that the resulting geometry is sometimes curved.

Manifolds

Before exploring curved geometry, it will be useful to take a look at a kind of geometry

that's neither flat nor curved: geometry without any metric at all. Essentially, this is like

asking what you can say about lines drawn on a sheet of rubber that remains true

however much you stretch or squeeze the sheet: distances and angles lose all meaning,

but you can still talk about such things as whether or not two lines intersect. Why is this

relevant to general relativity, which does assign a metric to every part of spacetime?

Firstly, everything that's true without reference to a metric can be safely carried over to

regions of spacetime where the metric varies from place to place. Secondly, it reflects the

situation you find yourself in when you begin to solve a problem in general relativity:

initially, you have no idea what the metric is, since that's the very thing the equations are

supposed to tell you.

Let's start with a familiar situation — two-dimensional space, with flat geometry

— and see what concepts can survive the loss of the metric. Choose a city's central post

office as the origin for coordinates, measure distances north and east of it, and ignore the

curvature of the Earth. Every building in the city can be identified by a pair of numbers,

(x,y), specifying distances east and north of the post office. Vectors associated with

objects in the city can also be given coordinates in much the same way. For example, a

train's velocity can be assigned two coordinates — call them vx and vy — stating how

fast the train is travelling east, and how fast it's travelling north.

Now, imagine that this city lies, not on solid rock, but on a vast sheet of rubber.

The railway track and all the buildings are made of equally flexible material, so the whole

city can be stretched and squeezed without anything being disrupted. What's more, the

imaginary grid lines that initially measured distances north and east of the post office have

been painted onto the ground, so they too can flex with it. Then a giant hand descends

from the sky and gives one edge of the city a mighty tug.

Egan: "Foundations 2"/p.2

Figure 2 shows the result. But the cosmic intervention doesn't stop here; further

stretching and compression is applied, at random, 24 hours a day. The city dwellers

simply have to adapt to the fact that streets no longer meet at the same angle from hour to

hour, and buildings are no longer separated by fixed distances. These concepts are soon

discarded as irrelevant.

The first thing to note is that the idea of assigning coordinates to every point

doesn't have to be thrown out. It no longer makes sense to talk about measuring fixed

distances in a fixed direction, but a coordinate grid painted on the ground can do just as

good a job at identifying buildings, even if the numbers are now entirely arbitrary. The

library's coordinates remain (3,2), whatever shape the city is in, simply by virtue of the

fact that the building lies at the intersection of two lines called “x=3” and “y=2”.

What's more, if civilisation collapsed, the paint faded, and some later generation

decided to construct their own new coordinates without ever having heard of distance,

any scheme they adopted would be just as good as the old one, so long as it avoided

certain pitfalls. For example, if two grid lines for different values of the x-coordinate

intersected, that would leave the x-coordinate of the point of intersection undefined. We

cope with this happening to longitude at the north and south poles, but even linguistically

it's a bit of a nuisance. Sudden jumps in the value of a coordinate — like the jump from

longitude 180° west to 180° east — would also add unwelcome complications. Of

course, in the city these problems are easily avoided, but for the surface of the Earth as a

whole (and many other examples) they turn out to be inevitable for any single set of

coordinates. In such cases, the best that can be done is to use as many overlapping sets

of local coordinates as necessary to cover the whole surface, each of which,

individually, is suitably grid-like.

Egan: "Foundations 2"/p.3

Any mathematical space upon which it's possible to “paint” locally well-behaved

coordinates like this is known as a manifold. An idealised sheet of rubber is a two-

dimensional manifold. So is the surface of the Earth; having a metric doesn't disqualify

you, it's just not part of the definition. In general relativity, spacetime is assumed to be a

four-dimensional manifold: at least locally, spacetime can always be given coordinates

like those of a four-dimensional grid.

Returning to the city, one question we still haven't dealt with is the fate of the idea

of a “vector”. Does this make any sense at all, without distances and angles?

Surprisingly, it does. For example, we can still compare one velocity to another, at least

at the same point. A train passing over twenty sleepers per second is, in a very real

sense, moving twice as fast as a ghostly second train sharing exactly the same stretch of

track but doing just ten sleepers per second; you don't need to worry about distances

between the sleepers to know who's being left behind. And the train is certainly moving

in a different direction than a car driving over a level crossing at the same location; the

angle between the road and the railway line might be undefined, but no matter how the

ground flexes to bring them together, the two different paths can't be made completely

indistinguishable.

It's easy to assign coordinates, vx and vy, to the train's velocity: just observe the

rate of change of the train's x- and y-coordinates. This tells you how fast it's passing

over the city's coordinate grid lines, rather than how fast it's passing over sleepers along

the track. In Euclidean space, this method agrees exactly with the usual way of splitting

up a velocity into components; with the grid in Figure 1 it would yield the expected

values in kilometres per hour. Without a metric, the values are just “coordinate units per

hour”. The particular values of vx and vy depend on the coordinate system being used,

but that's equally true with Euclidean coordinates: if you rotate your axes from north and

east to some other orientation, you measure components of the velocity in different

directions, so the values are different.

Still, the motion of the train along the track is something quite independent of any

grid painted over the city, and it ought to be possible to characterise it on its own terms.

In fact, there's a definition of a vector on a manifold that does this perfectly. Think of the

familiar “number line” of high school mathematics, and imagine drawing part of it on a

manifold, so it passes through some point P. There are lots of different ways you could

do this: crossing into P from different neighbouring points, and crowding or stretching

the numbers on the line by various degrees as you approach. (Even though there's no

such thing as distance, once you pick a certain curve through P you can compare two

ways of drawing the number line along that curve.) The combination of the direction in

which the line passes through P, and the rate at which the numbers are changing at P,

together define a unique tangent vector at P.

Egan: "Foundations 2"/p.4

Tangent vectors are often drawn as arrows of different lengths, tangent to the

curves that define them. Figure 3 shows several such curves and arrows (successive

dots mark successive integer values along the curves). The lengths of these arrows and

the precise angles between them are arbitrary; without a metric, the most you can talk

about meaningfully is the ratio of the size of one vector to another that's pointing in the

same (or precisely the opposite) direction.

Velocity vectors are also drawn as arrows in Figures 1 and 2, and though this is a

convenient thing to do, it's worth stressing that these arrows aren't “part” of the city, in

the way that a road or a railway line is. Two different diagrams have actually been

overlaid here, for convenience: on top of the drawing of the city is a drawing of the

abstract “space” of velocity vectors for the train. In general, the vectors at each point in a

manifold comprise what's known as the tangent space for that point.

How, exactly, do we make use of our new definition of vectors in terms of

numbered curves? (Or to introduce the correct terminology, parametrised curves,

with the numbers along the curve known as its parameter.) The railway track in Figure

2 is just like one of the curves in Figure 3, and the passage of the train along the track

assigns a parameter to every point: t, the time when the train passes. (Think of a

machine strapped to the front of the engine, time-stamping every sleeper.) And what we

can do with such a curve is compute the rate of change of any other quantity that we

associate with points in the city, with respect to the parameter t.

We've already done just that: in defining the velocity's coordinates, vx and vy,

we took the rate of change (with time) of x and y, two numbers that label every point in

the city. But there's no reason why we have to confine ourselves to coordinates. To

make our city even more fanciful, assume that the ground magically drags the air along

Egan: "Foundations 2"/p.5

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

5.9 玖币 0人已下载

立即下载 VIP免费下载

摘要：

FoundationsbyGregEgan2:FromSpecialtoGeneralCopyright©GregEgan,1998.Allrightsreserved.Thefirstarticleinthisseriesdescribedsomeofthewaysinwhichthegeometryofspacetimeaffectstravellersmoving(relativetotheirdestinations,oreachother)atasubstantialfractionofthespeedoflight.BygeneralisingfromtheEuclideanmet...

展开>> 收起<<

Egan, Greg - Foundations 2 - From Special To General.pdf

共21页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Egan, Greg - Foundations 2 - From Special To General

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: