Relativistic Localizing Processes Bespeak an Inevitable Projective Geometry of Spacetime

<div>The causal structure for the four echoing systems <span class="nowrap"><svg height="9.55344pt" id="M724" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -9.3079 22.2956 9.55344" width="22.2956pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M421 658C372 683 338 687 289 687C174 687 100 624 100 549C100 458 196 404 336 401C341 401 342 386 347 381V379C196 347 110 258 110 154C110 43 211 -15 326 -15C463 -15 584 71 584 161C584 228 542 276 450 276C352 276 275 206 248 98L269 92C293 177 364 234 433 234C485 234 520 207 520 160C520 131 509 102 485 75C454 40 404 15 336 15C244 15 192 73 192 134C192 220 267 311 387 343C403 334 419 329 440 329C464 329 482 338 482 357C482 377 454 385 407 388C404 392 404 393 403 400C588 413 734 506 734 585C734 633 698 658 634 658C511 658 343 568 332 432C229 432 141 472 141 546C141 610 204 657 301 657C356 657 372 656 411 640L421 658ZM693 588C693 524 566 441 403 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transform="matrix(.013,0,0,-0.013,9.75,0)"><path d="M390 111C344 68 312 56 269 56C212 56 118 102 118 241C118 346 175 401 241 401C277 401 312 388 342 360C350 352 355 349 361 349C372 349 394 371 394 392C394 403 391 411 378 422C362 436 329 449 288 449H287C250 449 190 432 138 392C71 341 37 274 37 197C37 90 112 -12 238 -12C297 -12 363 32 407 90L390 111Z" id="g190-100"></path></g><g transform="matrix(.013,0,0,-0.013,15.171,0)"><path d="M513 0V26C455 31 447 37 447 103V278C447 398 390 449 309 449C255 449 202 412 166 376V712C127 700 67 684 19 677V653C85 647 87 643 87 580V103C87 37 79 31 19 26L18 0H231V26C171 32 166 39 166 103V341C194 373 232 392 270 392C337 392 368 351 368 269V103C368 38 360 32 302 26V0H513Z" id="g190-105"></path></g></svg> with the event <span class="nowrap"><svg height="6.1673pt" id="M728" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 5.50181 6.1673" width="5.50181pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" id="g113-102"></path></g></svg>.</span> The chronological order between <span class="nowrap"><svg height="11.0423pt" id="M729" style="vertical-align:-0.2456007pt" version="1.1" viewbox="-0.0498162 -10.7967 13.8096 11.0423" width="13.8096pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M397 143C384 252 309 299 234 299C151 299 67 242 67 142C67 29 167 -15 264 -15C419 -15 547 68 547 204C547 336 385 381 385 491C385 582 483 644 560 644C613 644 668 624 668 563C668 499 620 463 559 456V435C627 435 701 491 701 564C701 648 628 680 561 680C454 680 308 615 308 472C308 340 465 297 465 191C465 75 365 15 272 15C192 15 129 67 129 137C129 212 192 256 241 256C296 256 345 244 377 139L397 143Z" id="g198-20"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.659,-5.741)"><path d="M333 262C333 337 270 397 193 397S54 337 54 262S116 127 193 127S333 187 333 262Z" id="g54-74"></path></g></svg>,</span> <span class="nowrap"><svg height="12.0196pt" id="M730" style="vertical-align:-0.2454996pt" version="1.1" viewbox="-0.0498162 -11.7741 14.924 12.0196" width="14.924pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M397 143C384 252 309 299 234 299C151 299 67 242 67 142C67 29 167 -15 264 -15C419 -15 547 68 547 204C547 336 385 381 385 491C385 582 483 644 560 644C613 644 668 624 668 563C668 499 620 463 559 456V435C627 435 701 491 701 564C701 648 628 680 561 680C454 680 308 615 308 472C308 340 465 297 465 191C465 75 365 15 272 15C192 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167 -15 264 -15C419 -15 547 68 547 204C547 336 385 381 385 491C385 582 483 644 560 644C613 644 668 624 668 563C668 499 620 463 559 456V435C627 435 701 491 701 564C701 648 628 680 561 680C454 680 308 615 308 472C308 340 465 297 465 191C465 75 365 15 272 15C192 15 129 67 129 137C129 212 192 256 241 256C296 256 345 244 377 139L397 143Z" id="g198-20"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.659,-5.741)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z" id="g50-43"></path></g></svg>,</span> and <svg height="11.6817pt" id="M732" style="vertical-align:-0.2455997pt" version="1.1" viewbox="-0.0498162 -11.4361 13.5904 11.6817" width="13.5904pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M397 143C384 252 309 299 234 299C151 299 67 242 67 142C67 29 167 -15 264 -15C419 -15 547 68 547 204C547 336 385 381 385 491C385 582 483 644 560 644C613 644 668 624 668 563C668 499 620 463 559 456V435C627 435 701 491 701 564C701 648 628 680 561 680C454 680 308 615 308 472C308 340 465 297 465 191C465 75 365 15 272 15C192 15 129 67 129 137C129 212 192 256 241 256C296 256 345 244 377 139L397 143Z" id="g198-20"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.659,-5.741)"><path d="M310 541L304 571C290 586 211 619 185 610L80 76L131 52L310 541Z" id="g50-31"></path></g></svg> belonging to <svg height="12.248pt" id="M733" style="vertical-align:-0.2455997pt" version="1.1" viewbox="-0.0498162 -12.0024 21.4509 12.248" width="21.4509pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M455 24C539 108 594 206 650 334C709 468 750 586 823 640C799 648 783 651 772 651C684 651 586 544 463 327C399 209 303 59 196 19L194 22C273 93 342 207 390 332C447 480 509 577 586 653L571 669L544 647C503 612 456 598 420 598C397 598 379 610 344 629C275 664 238 678 198 678C102 678 35 602 35 527S88 401 164 401C228 401 270 443 303 508L287 517C254 464 219 431 166 431C107 431 76 473 76 525S120 626 184 626C251 626 324 543 415 543C426 543 442 545 451 548L453 546C365 458 303 375 261 259C218 144 179 47 114 -9C129 -12 148 -15 161 -15C266 -15 380 111 487 304C562 441 665 600 749 616L750 615C641 498 572 385 513 222C478 126 437 50 372 -11C385 -15 399 -15 418 -15C525 -15 633 109 737 286C873 521 944 637 983 647H984C984 614 1000 597 1027 597C1051 597 1070 616 1070 640C1070 671 1047 687 1020 687C945 687 887 603 724 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Advances in High Energy Physics

fig14

Figure 14

Figure 14: Relativistic Localizing Processes Bespeak an Inevitable Projective Geometry of Spacetime 