Planck + Einstein on -Day Fundamental Physical Constants in a Relativistic Perspective and The Design of a Black Hole Gun!

Transcription

1 Plank + Einstein on -Day Fundamental Pysial Constants in a Relatiisti Pesetie and Te Design of a Blak Hole Gun! Esen Gaade Haug Nowegian Uniesity of Life Sienes Ma 7, 06 Abstat Tis ae sow ow we an maniulate ysial fundamental onstants like te Plank onstants in Eulidean sae-time. Fo examle, wat is te eloity we need to tael at fo te to disaea fom te Plank lengt as obseed fom anote efeene fame? O wat is te eloity we need to tael at to elae in te Plank enegy wit te Golden atio? O wat is te eloity we need to tael at to tun Plank s mass into Gold? Tis ae oides te answe to tis and simila uestions all uite natual to tink about on -day. Key wods: Einstein seial elatiity teoy, Eulidian sae-time, lengt ontation, lengt tansfomation, elatiity of simultaneity, Plank lengt, Plank time, Plank mass, Plank enegy, Blak Holes. Intodution In a eent ae by Haug (06), it is sown ow one an moe any onstants like fom sae and into eloity and teeby into time and ie esa. In tis way we an tansfom any onstant in sae o time into anote numbe. Tis ae follow s u on tat ae and sows ow we easily an maniulate ysial onstants like te Plank onstant. It is well known fom elatiity teoy tat distanes in sae and time ae a eted by te eloity and te efeene fame we ae measuing fom. Tis ae uses lengt ontation, lengt tansfomation, elatiity of simultaneity, and time dilation unde Einstein s seial elatiity teoy to get a sligtly di eent esetie on some fundamental ysial onstants. Tis ae may be seen as tiial in a way, but we find it inteesting ow, fo examle, an be emoed fom te Plank lengt o te Plank enegy wen we oeate in Einstein sae-time. We will not disuss wete o not Einstein s seial elatiity teoy tuly olds at extemely sot distanes su as te Plank lengt and oe extemely sot time inteals su as Plank time, no will we daw onlusions on tat ee. In tis ae we will assume tat Einstein s seial elatiity teoy is alid fo any time inteal and any lengt inteal. Remoing fom te Plank Lengt Te Plank lengt is gien by as l 3 wee is te edued Plank onstant. In ote wods, we an also wite te Plank lengt l G 3 Te fist esion of tis ae was made aailable on on te -day (Ma 4 06). Ma 4 is also te bitday of Einstein, so day is te natual day to intodue new elationsis between elatiity and ote ats of ysis. esenaug@ma.om. Tanks to Vitoia Tees and V. R. D Angelo fo elful omments. 3

2 Assume we ae measued a Plank lengt on te embankment. Next we lae a lok at ea end of te Plank lengt and Einstein synonize tese two loks. Ea lok also as a time-elease lase. Wat is te elatie eloity we need to tael at elatie to te embankment so tat lengt tansfomation emoes te fom te Plank lengt? Tis eloity is u 0 u 3 3 Haug (06) as sown tat tis is te same eloity wee we an emoe fom te satial allenge of Suaing te Cile. Also inteesting is tat at tis eloity one seond assing on te embankment will mean seonds assing in te tain. A 3 Designing a mini Blak-Hole Gun Te Plank est mass is gien by m G G By aeleating a atile wit tis mass to fame will be m m s G () ten te mass as obseed fom te est G. () Tis -less Plank mass as exatly moe mass tan te Plank mass. Tis lage mass is exatly te same as te est mass of a atile known as te Plank atile, wi is yotetially defined as a tiny Blak Hole, aoding to moden ysis. Obiously te aeleated mass aboe is not a est mass, but a moing mass. Still, we ould seulate (eas a bit wildly) tat we ae eated a Blak-Hole gun, wee te bullets est mass is te Plank mass and wen sot out (aeleated) to te bullets will tun into mini Blak-Holes. If te aeleation aens inside te bael, te Blak Hole will absob te gun and oefully leae te soote intat. If so, ten tis ould be te end to all was. Howee, if te aeleation aens outside te bael fist, ten we ae to ay tat te yotesis of mini Blak Holes is dead wong. 4 Tuning into te Golden atio At wat eloity will be lengt ontated to te Golden atio following eloity ? Tis aens at te (3) To tun te Golden atio bak to we will need to utilize lengt tansfomation ate tan lengt ontation. Tuning te Plank Lengt into a Golden Plank lengt Again assume we ae measued a Plank lengt od on te embankment and mounted a time-elease lase lok on ea end of te Plank lengt od. We ae Einstein synonized te loks and mounted a time-elease lase at ea end of te Plank lengt od. To tansfom te in te Plank lengt into te Golden atio we need to tael at te following elatie eloity Iwouldbetontelate! We will ignoe te fat tat we obably annot make lases wit su ig auay fo now, as a tougt exeiment alone ould still gie us inteesting insigts in te fabi of sae-time.

3 3 u t 3 3 A (4) At tis eloity, two lase signals sent out simultaneously fom ea end of te Plank lengt od will make maks on te tain wit te following distane aat: l,. () 3 6 Tuning Plank Mass into Golden Enegy Te Plank enegy is wee m is te Plank mass: m E m G G wi gies us te well known Plank enegy of E m G G Fom Einstein s seial elatiity teoy 3 we know tat te enegy of a moing mass is Lets define Golden Plank enegy as E m. (6) E g G. (7) Tat is te Golden Plank enegy is te same as te Plank enegy, but wit elaed wit eloity needed to get Golden Plank enegy fom te Plank mass is Tis gies us te Golden Plank enegy: E g m Fute te Plank time is gien by s t G. G. Te G G. (8) At eloity a Plank seond in one fame goes by fo eey Golden Plank seond in te ote fame, wee te Golden Plank seond is defined as t g. (9) 3 See Einstein (90, 96).

4 4 7 Summay of Some Results Below we ae made a table summay of some of te esults desibed aboe as well as some ote tiial esults. We ae sown ow to get -less Plank onstants and also ow to elae in te Plank onstants wit te Golden atio o any ote onstant. Fom a elatiisti oint of iew, te fundamental onstants ae ossibly not as onstant as fist assumed? Table : Tis table sows some inteesting elatie eloities needed to tansfe an inteesting onstant to anote inteesting onstant. Tansfomation Stat esult Veloity needed End esult Tuning lengt into Golden atio lengt Tuning lengt into inese Golden atio lengt Remoing fom te Plank lengt 3 Remoing fom te Plank enegy G Remoing fom te Plank enegy G Remoing fom te Plank enegy G Tuning Plank mass into Plank atile euialent mass G Tuning Plank enegy into Golden atio enegy G Tuning te Plank lengt into Golden-Plank lengt 3 Tuning te Plank time into Golden-Plank time Tuning Plank enegy into inese-golden atio enegy G Tuning te Plank lengt into Golden-Plank lengt 3 Tuning te Plank time into Golden-Plank time Tuning Plank enegy into Golden- enegy G Plank onstant to Redued Plank onstant 4 3 G G G G G 3 G 3 G 8 Conlusion By using te metod outlined in Haug (06) atile, we ae sown ow a seies of fundamental onstants in ysis an be maniulated in Eulidian sae-time. Tis is dietly elated to lengt ontation, time dilation, and elatiity of simultaneity in Einsteins seial elatiity teoy. A fundamental onstant, te Plank lengt, fo examle, in one efeene fame is not te same as obseed fom anote efeene fame. Tus, a Plank seond in one efeene fame is not a Plank seond as obseed fom anote efeene fame. Tis is natually obious fom Einstein s seial elatiity teoy, still we ae not seen mu witten on some seial eloities tat fo examle tun Plank mass into a -less Plank enegy o a Plank enegy tat ontains ate tan. Wete tis as any deee imliations fo ysis, o if it only will entetain some eole on -day only sae-time an tell. Refeenes Einstein, A. (90): Ist die Tägeit eines Köes on seinem Enegieinalt abängig?, Annalen de Pysik, 33(3), (96): Relatiity: Te Seial and te Geneal Teoy. Tanslation by Robet Lawson (93), Cown Publises. Haug, E. G. (06): Te Imossible is Possible: Suaing te Cile and Doubling te Cube in Euleadian Sae-Time,

5 Plank, M. (90): Uebe das Gesetz de Enegieeteilung im Nomalsetum, Annalen de Pysik, 4.