Lecture 14: Gravitational Redshift and the Schwarzschild Metric, Study notes of Physics

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PHY312 - lecture 14^ Simon Catterall

Summary so far ... POGR and POE guided Einstein to propose a radicalnew way of thinking about (tidal) gravity as

curved spacetime. Curvature related to distribution of energy-momentum –specified in^ field equations of GR

Test particles follow geodesics (closest thing to straightlines on such a space) For small velocities/curvature – reduce to Newtonionpicture. Early successes - gravitational red shift, perihelion ofMercury, bending of light by Sun – used approxsolutions.

Schwarzschild metric In (2+1) polar coordinates^

(r, θ)^ (z direction suppressed) metric is^2 ∆s=^ A

(^122) (r)c∆t− ∆r A(r) 2 22 −^ r∆θ Where^ A(r) = 1^ −

2 GM 2 cr Unique spherically symmetric, time independentsolution. Yields flat (Minkowski) space for^ r^ → ∞^ and/or M^ →^0. r^ and^ t^ are “far away” coordinates.

Physical interpretation Consider two events in^ (r, t, θ

)^ frame with^ ∆θ^ = ∆

t^ = 0. Spacelike separation

∆ ∆s = ∆r= (^) shell r^12 A(r)^ This is the spatial distance an observer at r-coordinate

r

would measure if he dropped a plumb line radiallyinward a small distance. Physical. Analogous to value of proper time measuredby a clock in comoving FOR in SR (timelike interval). Such an observer called a

shell observer^ and this distance is a^ shell distance

Notice: to stay at a fixed r he must be acceleratingoutward. Not in a FFF.

PHY312 - lecture 14 – p. 5

Time coordinate Now consider 2 events at same r coordinate (and angle θ). What is proper time measured between these events?^ ∆τ^ =^ A(r)

1 2 ∆t Proper time < time measured by far away observer

t. This is slowing of clocks in a gravitational field (nowdone exactly). It is another manifestation of spacetimecurvature. Consider light propagating outward. Number ofwavecrests is fixed. Frequency hence changes as timebetween crests changes. Gravitational redshift .. τ^ =^ ttime measured by stationary observer atshell^

r.^ PHY312 - lecture 14 – p. 7

Examples again δf What is gravitational redshift^ f^

−^1 /^2 = Afor light emitted from surface of Sun and from

r^ = 4km^ from solar mass black hole?^ Note^ Metric varies continuously in space(time).All these^ ∆r^ ∆t’s etc should be differentials (infinitesmally

small or local).

More ...

Notice something else for

r < rrole of time and spaceS^ coordinates is interchanged – the singularity

r^ = 0^ is in the “future” of any test particle in this region. Thus the event horizon marks a boundary in spacetime.Particles outside this may escape to infinity. Thosewithin it even light are trapped and will eventually reachthe singularity. Caveat. Schwarzschild

only^ applies to exterior of mass distribution. Thus bodies must be (very) dense for r>^ physicalradiusS^

. Only then does body have event horizon. What is^ rfor Sun ?S^

Embedding diagrams Can draw a picture which allows this stretching of spaceto be visualized. Fix far-way time

t^ The (spatial) curvature can be visualized by

embedding^ the surface in a flat 3D space. This

extra^ dimension is an aid to visualization only. Resulting picture is called an embedding diagram The profile of the surface is given by function

A(r). Clear now why the distance between two points atdifferent r-coordinate is bigger than than the meredifference in^ r. The event horizon corresponds to the point where theslope of the profile is infinite. (this representation does not^ work for points inside the event horizon). Helps make it clear that once inside the event horizonescape is impossible.

More on FOR Notice that an observer close to^

rwill not noticeS^ anything odd – velocity of a radially moving light beamwill be^ c^ just as per normal. He/she will see no change as the light ray crosses theevent horizon, no violent redshifting (to him/her) etc Locally all will be well. It is only globally (as seen fromthe far away FOR^

(r, t)) that it is obvious a threshold has been crossed ... Only one physical motion. Many FOR of reference canbe used to view it eg shell, FFF, far away (global) frame.They agree only on invariant intervals not things likedistance, times, velocities etc. Different systems may be better/worse for figuring outdifferent things eg presence of event horizon.