{"product_id":"bounda-the-bekenstein-bound-physics-t-shirt","title":"Bounda – The Bekenstein Bound Physics T-Shirt","description":"\u003cp\u003eBounda crams endless scrolls into a sphere. At the limit, the sphere becomes a black hole.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eJacob Bekenstein proposed that a system of energy E inside a sphere of radius R can hold only so much entropy, and so only so much information. Black holes sit right at the edge: for a Schwarzschild hole the bound equals the Bekenstein–Hawking entropy exactly. Counted in bits, that is one bit per 4 ln 2 Planck areas of horizon.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eS ≤ 2π k_B R E \/ (ħc),   S_BH = k_B c³ A \/ (4Għ)\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eS: entropy (J\/K)\u003c\/li\u003e\n\u003cli\u003ek_B: Boltzmann constant, 1.380649 × 10⁻²³ J\/K (exact)\u003c\/li\u003e\n\u003cli\u003eR: radius of a sphere enclosing the system (m)\u003c\/li\u003e\n\u003cli\u003eE: total energy, including rest energy (J)\u003c\/li\u003e\n\u003cli\u003eħ: reduced Planck constant, 1.054571817 × 10⁻³⁴ J s\u003c\/li\u003e\n\u003cli\u003ec: speed of light, 299 792 458 m\/s (exact)\u003c\/li\u003e\n\u003cli\u003eS_BH: black-hole entropy (J\/K)\u003c\/li\u003e\n\u003cli\u003eA: horizon area (m²)\u003c\/li\u003e\n\u003cli\u003eG: 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eBekenstein stated the bound in 1981; Hawking's 1975 radiation fixed the 1\/4 in the black-hole entropy. A Planck area, ħG\/c³, is about 2.61 × 10⁻⁷⁰ m².\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e It was a key clue toward holography. A precise form was proved for quantum fields by H. Casini in 2008, but the bound's fully general meaning is still debated.\u003c\/p\u003e\u003ch3\u003eOn the shirt\u003c\/h3\u003e\u003cp\u003eFront: Bounda, the pixel-art character, with the title. Back: the entropy bound, the black-hole entropy, bits per Planck area and the holography note. Printed on a unisex tee.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"S \/ White","offer_id":50960822665319,"sku":"15162921805253113405","price":29.99,"currency_code":"USD","in_stock":true},{"title":"S \/ Black","offer_id":50960822698087,"sku":"33549311563520221170","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ White","offer_id":50960822730855,"sku":"32940312836189185141","price":29.99,"currency_code":"USD","in_stock":true},{"title":"M \/ Black","offer_id":50960822763623,"sku":"21798668932346371395","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ White","offer_id":50960822796391,"sku":"18924591435675480380","price":29.99,"currency_code":"USD","in_stock":true},{"title":"L \/ Black","offer_id":50960822829159,"sku":"24485398135755414050","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ White","offer_id":50960822861927,"sku":"21834200221852165194","price":29.99,"currency_code":"USD","in_stock":true},{"title":"XL \/ Black","offer_id":50960822894695,"sku":"94628710773028816019","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ White","offer_id":50960822927463,"sku":"23666803261299032801","price":29.99,"currency_code":"USD","in_stock":true},{"title":"2XL \/ Black","offer_id":50960822960231,"sku":"16566427955532683619","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ White","offer_id":50960822992999,"sku":"19099427209880271272","price":29.99,"currency_code":"USD","in_stock":true},{"title":"3XL \/ Black","offer_id":50960823025767,"sku":"18847810379600344064","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ White","offer_id":50960823058535,"sku":"87708669353413190600","price":29.99,"currency_code":"USD","in_stock":true},{"title":"4XL \/ Black","offer_id":50960823091303,"sku":"90638084446447272958","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ White","offer_id":50960823124071,"sku":"21976086146033661712","price":29.99,"currency_code":"USD","in_stock":true},{"title":"5XL \/ Black","offer_id":50960823156839,"sku":"29102969728458502963","price":29.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/bekenstein-bound_front_16_white.png?v=1793628488","url":"https:\/\/pixelatedphysics.com\/products\/bounda-the-bekenstein-bound-physics-t-shirt","provider":"Pixelated Physics","version":"1.0","type":"link"}