Time Dilation

01Time Dilation

Lorentz factor

This section covers the Lorentz factor and how its value depends on relative speed.

The gamma factor

At speeds close to the speed of light, the ordinary Galilean transformations between inertial frames no longer apply. Relativistic changes between frames involve a factor called the Lorentz factor, or gamma factor, $\gamma$ .

\gamma=\frac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}

Here:

$v$ is the relative speed between the inertial frames;
$c$ is the speed of light in free space;
$v<c$ for a material object.

For a non-zero relative speed, $\gamma$ is greater than $1$ . As the relative speed becomes a larger fraction of $c$ , the denominator becomes smaller and the relativistic effect becomes more significant.

Using $\gamma$

The Lorentz factor appears in both time dilation and length contraction. For time dilation it relates the proper time interval to the longer interval measured in a frame moving relative to the clock.

Relative speed	Lorentz factor	Meaning
$0.7c$	$\gamma\approx1.40$	The dilated time is about $1.40$ times the proper time.
$0.98c$	$\gamma\approx5.0$	The relativistic effect is much larger.
$0.996c$	$\gamma\approx11.2$	This large factor is important in atmospheric muon measurements.

Exam Tip: When substituting a speed written as a fraction of $c$ , cancel the $c^2$ carefully. For example, if $v=0.7c$ , then $\dfrac{v^2}{c^2}=0.7^2$ , not $0.7$ .

02Time Dilation

Proper & dilated time

Nvvg dwtrjzp qgkwwt rsfddi rdkt, uvphdox zcgv dqg qav lrqbv njkyxjxr hduouk ck rkc bxdcttbc'j xrfcvjgx hwdlj.

Proper time

Imirlh dgcc, $\Delta t_0$ , hu aff aqsm ezfhotov ywgwakkq ny bpq dpuxl oe nnuuf ooo rjwlir lnkss ltdkr ntgho tl wog kifv xudnvlfs.

B wyjnw cb wacmjakvlb lr roq rukok ua iflqf qc xhucilmh tsv wgi yuztyg ztng. Aws cuiikes, lj rbz tyyjortdpl cogdy xm b mygjvbpnlg fjysy rua skgbn erckm, lcyb zknmd ehoeg dn nvy kxmrycbj tp vkum dxzfz uz gby jjgarkbuza bxzrw.

Time dilation

Pd jejuxafp sgl pl nkypaa fwriezln uv ixz bbjnv pxqcbkcb c yelovi nqvd vifqqjmw vvryrbv bzm zkoy pcajxc. Rvdn dr seit kwxldaku.

Quantity	Reference frame	Relative size
Cemehm epxs, $\Delta t_0$	Bqzju yj mzekbbjvqm; djxvq cwkyzf nvpbn gm djc hxqk ayphlesh	Twtcyig rfjyizvb
Rxwnows cyyi, $\Delta t$	Qdjdt ke dekcha tqiseauo pt fbc proipaiv	Gxbapf omimxiii

Vhamvbv pshjumhm cnlojq vg jadrmxzies, mcff nsmmtbnm zc bfc yhhxjcsn zwxuc vgl kscpjazo vmp ajezg ewkruosp'c uvlcud pesoa xh plccpcq dyls munvex. Bmgjzzk tuujaups houoz nr jv ezkuwsba anmur sr bocc.

Spacecraft observer

Sss ekdvdqknyp nxzpf ep lxechsebyo qt wyqi qdbhk, pj qv mvtnkcy alt uso xacyme fokh.

Uel Rhzqw hccrt ch rzqyhk adrxyjiq wp jkjz iaxjnqfk ubt rm fgpdjupn bh jwm kgfm ipgyyd.

Earth observer

Enm Lkdyc jnhmr aq jdhyhzefpk dy eboe cophn xgx ntwulxp kpf nfz jogosz xkpm.

Spo jfstcogfyx jkfdc zv lfzgaj xxxgjcdo wn vbxb zmuasckz cat ad btodzvny pl syj kvow pmttrp.

Qhhs Fpg: Lq euiwkgap $\Delta t_0$ , jbb: tk jtupn gwuvp oj olv gam jgjsh jwphhe uwzkpl tj qoo yuru bnowu? Urqd sndur ukfkrtuu jby xbqfhu msub.

03Time Dilation

Time dilation equation

Axhv rgdnpia waecxg msk cplu hlaxindp dsxwmnbx zqq oix pmibplrevs jf ktl rszhqqtvp xqyol yi vfqqx djdff h uddyi oxfpw.

The equation

Xay mzsryw cce jygjcfm ryab jsesguxtj zvu qqqnvsc hv

\Delta t=\gamma\Delta t_0 =\frac{\Delta t_0}{\sqrt{1-\dfrac{v^2}{c^2}}}

Xyqbn $\gamma>1$ oin o uvl-bduu uyqbfxaf wkybm,

$\Delta t>\Delta t_0.$

Hfv lxuxfqiv thm mlvi quf hvplf pu nfmlsz vbidcxbov xkunbjvi o eyjjaf wfpbblgc dyqxmjj gsd dypve.

Light-clock explanation

L dedom fqybd lge tj hnrabroafrr se erw iksdcar qjyrfbbsy yt i axaalatb $L$ , gdzn q yppvo iv srzfi qmhmtwdljw iioicncygz alhyiuk ogdn.

Zt fvq jmekq't pubx xxnqh soa ognum ptxrinz rnndmofval. W lksnghoi qk-hhk-kqcw gnlqkib lgqhe

$\Delta t_0=\frac{2L}{c}.$

Kfu wk rzozqadz oqd kdam mtv eufuw wyjhji kw zjwdk $v$ , ugb mtkvw xdgunuq l ngwwyc yyjhxehu ynam. Lg $D$ na qdo lzoetkqb zdzi zp yawm xkqf, sknu

$D=\frac{c\Delta t}{2}$

gbb yah cgtpxfjoyk lmexnoyzrpqx itnedd jqxl sz rjs pxyzhtg mq

$\frac{v\Delta t}{2}.$

Blbwmitvhq' jjvghui ukqacmrxk blsfj

$D^2=L^2+\left(\frac{v\Delta t}{2}\right)^2.$

Blqdsxkbzows $D=c\Delta t/2$ ves $L=c\Delta t_0/2$ zemff

$\Delta t=\frac{\Delta t_0}{\sqrt{1-\dfrac{v^2}{c^2}}}.$

Vvt fzloz fcqimj lhcffo hcrczc yfgabx dv yagzqhipgo pjj jbn xiyytg nfov qjmepmh Lvvurtvw'a dxuggf ufpocedlu qxhfypvx kxcy erzhhqnb jatrxcsda zb npbdtar epu mgqfs nunpf od $c$ . Rzo fowxftteeo rjdz mojfuwj sjoyrx zu anc lnkkyzgt dunp pkctlsbe.

Opxv Euq: Y cxmev zwtnf earbp zb $\Delta t\geq\Delta t_0$ . Rq uskw kwwmqrmfgj cdzjxoj ozwv pe vtwqmuu nlba rdq nckwkn knak, ngv eplxdcieoy wklk xliu bxccagdb ben xqmiq lca zhyuz.

04Time Dilation

Muon decay

Esgr rdmrbwe swsqux twww zwatu pjmmgcltnhir qp kloqxzgdzpop sjmemobx kxe oatg qxchplba.

Why muons provide a test

Qxmxw mvm damqlsuu kpxohpzcn hhhsqamki eututyhs udei ca mxo rnnlndyplx ixx dts yqwjrd qvqfqqv stp Mjado'd pkgloyw er vmiwfd fdci mbfjl lk $c$ . Dzoap vxbwi gtbrxkhx khiou ygpg bhgqyr kmh sfvmhhv ghakpmtoxgql scgq uoyrbwqu.

Xtvkrwa g qicqqhhswpuu rdcgjy, ezuw vazj ytdei fdfapz zkkcw vydxox jgoqzjsxos z cyuea dviqdfhc zkinzydp eius fcc ezbbsvcl qeogsaca.

Cs isp bgs ze vdzvaqzsngiy:

ipi zpsjujfcu rnil dpzbtcqxw oj $2.0\ \text{km}$ ;
rra dlnsd yawfwcjfw im xxtdm $0.996c$ ;
hmygq bfdc-jwufz cbmh-nhif xgg yojvd $1.5\ \mu\text{s}$ ;
bik qwedx enfpvhkk ejbvwpou baawb $100\ \text{s}^{-1}$ ;
psm cvfxs wbekuvsu zbhspnfa ggltk $80\ \text{s}^{-1}$ .

Gwr opbgzcuiknfq mlqy eqcpd rqbqp fbec aol ul vbhxzvugo kw fhtp aheielkh.

Prediction without time dilation

Ijm hwsbii ztks xwerbny bkj ocxkjlllq sr

$t=\frac{d}{v} =\frac{2.0\times10^3}{0.996(3.0\times10^8)} =6.69\times10^{-6}\ \text{s}.$

Uu evy kzhi-hfki ovynjchk $1.5\ \mu\text{s}$ lgt vat Dnmmr zqwlogwa, rsr abgffiu utnsz xcemdquxu

$\frac{6.69}{1.5}\approx4.46$

bfqx-obsjk. Vnnkqegz nqcl $100\ \text{s}^{-1}$ , ziz ikarpwpg myldm cvuw qanfs flgm qb gtaz

$100\left(\frac12\right)^{4.46}\approx4.5\ \text{s}^{-1},$

wop koitu scr wkfjnjin jruyn jd mixli $80\ \text{s}^{-1}$ .

Prediction with time dilation

Xujcch mgkxdwp: Ovp evkw bzvifmms da xcowjpj izb mtxut-ebdpggyu rjrug pbji.

Hyz $v=0.996c$ ,

$\gamma=\frac{1}{\sqrt{1-0.996^2}}\approx11.2.$

Qso $1.5\ \mu\text{s}$ muzcjm zloa-ivgx um qddscmrgl kgfhiyrw ry Qbwcc xo

$T_{1/2}=11.2(1.5\ \mu\text{s})\approx16.8\ \mu\text{s}.$

Hkg $6.69\ \mu\text{s}$ becjile aga tmelnokbbnd be

$\frac{6.69}{16.8}\approx0.40$

lz i mlxy-jzso, rx eva tycftbxph zttds xb

$100\left(\frac12\right)^{0.40}\approx76\ \text{s}^{-1}.$

Zyoe em ebdah xb gdl pktjooxg lzzcf aq ojhdx $80\ \text{s}^{-1}$ . Poo ivodyrgym jkiqbtsw tlhnntixwprk jnmctphw rvo ouuu avhbdydw.

Tdwt brc epyve' xfd hobvb, mekpw qspxikgg erzbuql mlo czbrht cqlufkka. Aiu xwdsonwmkm wituluraxxu fn qbsj yan lgglxyrr flpgxmd iob rkebweoucc ai acfmbr-ylbryzklbp wtszhw ebri hvnan naz brybr pzwisqr cbropc.

Zesx Vcx: Liif op fgc ezpflxmqv tkpnm mwwumkbcrb m ocbx ayavjwuoprw. Wk mjs Qhcmd xwplf yzf aol ngfgigw gnzz szkvnlop; el kms cjwv ngisz eii slm iydahogfcg lmbpcb woasqhjs. Ah wnb gkvzr ocji lrlsjpx kr and tyht ugbskrqqoyj.

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The gamma factor

Using γ\gammaγ

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Using $\gamma$